Probability Statistics and Random Processes for Electrical Engineering 3rd Edition Leon Solutions Manual
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Probability Statistics and Random Processes for Electrical Engineering 3rd Edition Leon Solutions Manual Full download : https://goo.gl/yUaJzf Probability Statistics and Random Processes for Electrical Engineering 3rd Edition Leon Solutions Manual A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineers 1-1 Chapter 1: Probability Models in Electrical and Computer Engineering 1.1 1.2 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineers 1-2 1.3 1.4 1.5 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineers 1-3 1.6 1.7 1.8 1.9 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineers 1-4 1.10 1.11 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-1 Chapter 2: Basic Concepts of Probability Theory 2.1 Specifying Random Experiments 2.1 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-2 2.2 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-3 2.3 2.4 2.5 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-4 2.6 2.7 2.8 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-5 2.9 2.10 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-6 2.11 2.12 2.13 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-7 2.14 2.15 2.16 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-8 2.17 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-9 2.18 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-10 2.19 2.20 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2.2 2-11 The Axioms of Probability 2.21 2.22 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-12 2.23 2.24 2.25 2.26 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-13 2.27 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-14 2.28 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-15 2.29 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-16 2.30 (c) 2.31 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-17 2.32 2.33 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-18 2.34 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-19 2.35 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-20 2.36 2.37 2.38 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2.3 2-21 *Computing Probabilities Using Counting Methods 2.39 2.40 2.41 2.42 2.43 2.44 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-22 2.45 2.46 2.47 2.48 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-23 2.49 2.50 2.51 2.52 2.53 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-24 2.54a (b) 2.55 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-25 2.56 2.57 2.58 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-26 2.59 2.60 2.61 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2.4 2-27 Conditional Probability 2.62 2.63a © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-28 2.63c 2.64 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-29 2.65 2.66 2.67 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-30 2.68 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-31 2.69 2.70 2.71 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-32 2.72 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-33 2.73 2.74 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-34 2.75 2.76 (b) © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-35 2.77 2.78 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-36 2.79 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-37 2.80 2.81 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2.5 2-38 Independence of Events 2.82 2.83 2.84 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-39 2.85 2.86 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-40 2.87 2.88 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-41 2.89 2.90 2.91 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-42 2.92 2.93 2.94 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-43 2.95 2.96 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2.6 2-44 Sequential Experiments 2.97 2.98 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-45 2.99 2.100 2.101 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-46 2.102 2.103 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-47 2.104 2.105 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-48 2.106 2.107 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-49 2.108 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2.7 2-50 *Synthesizing Randomness: Random Number Generators 2.109 2.110 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-51 2.111 2.112 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-52 2.113 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-53 2.114 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-54 2.115 2.116 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2.8 2-55 *Fine Points: Event Classes 2.117 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-56 (d) © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-57 2.118 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. 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For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Instructor’s Solutions Manual Probability, Statistics, and Random Processes for Electrical Engineers 2-64 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-1 Chapter 3: Discrete Random Variables 3.1 The Notion of a Random Variable 3.1 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. 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For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-10 3.16 3.17 3.18 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-11 3.19 3.20 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. 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For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-13 3.22 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-14 3.23 3.24 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. 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For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3.4 3-22 Conditional Probability Mass Function 3.35 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-23 3.36 3.37 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. 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This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-48 3.77 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3.6 3.78 3-49 Generation of Discrete Random Variables The following Octave commands will give the requested plots: (a) x = [0:1:10]; lambda = 0.5; figure; plot(x, poisson_pdf(x, lambda)); figure; plot(x, poisson_cdf(x, lambda)); figure; plot(x, 1-poisson_cdf(x, lambda)); x = [0:1:20]; lambda = 5; figure; plot(x, poisson_pdf(x, lambda)); figure; plot(x, poisson_cdf(x, lambda)); figure; plot(x, 1-poisson_cdf(x, lambda)); x = [0:1:100]; lambda = 50; figure; plot(x, poisson_pdf(x, lambda)); figure; plot(x, poisson_cdf(x, lambda)); figure; plot(x, 1-poisson_cdf(x, lambda)); (b) x = [0:1:15]; figure; plot(x, binomial_pdf(x, 48, 0.1)); figure; plot(x, binomial_cdf(x, 48, 0.1)); figure; plot(x, 1-binomial_cdf(x, 48, 0.1)); x = [0:1:30]; figure; plot(x, binomial_pdf(x, 48, 0.3)); figure; plot(x, binomial_cdf(x, 48, 0.3)); figure; plot(x, 1-binomial_cdf(x, 48, 0.3)); x = [0:1:50]; figure; © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-50 plot(x, binomial_pdf(x, 48, 0.5)); figure; plot(x, binomial_cdf(x, 48, 0.5)); figure; plot(x, 1-binomial_cdf(x, 48, 0.5)); x = [20:1:50]; figure; plot(x, binomial_pdf(x, 48, 0.75)); figure; plot(x, binomial_cdf(x, 48, 0.75)); figure; plot(x, 1-binomial_cdf(x, 48, 0.75)); (c) x = [0:1:10]; n = 100; p = 0.01; figure; plot(x, binomial_pdf(x, n, p), "1"); hold on; plot(x, poisson_pdf(x, n*p), "3"); hold off; © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3.79 3-51 The following Octave commands produce the request plots: (a) L = 10; k = [1:1:L]; cL = sum(1./k); pk = (1/cL)./k; figure; plot(k, discrete_pdf(k, k, pk)); figure; plot(k, discrete_cdf(k, k, pk)); figure; plot(k, 1-discrete_cdf(k, k, pk)); L = 100; k = [1:1:L]; cL = sum(1./k); pk = (1/cL)./k; figure; plot(k, discrete_pdf(k, k, pk)); figure; plot(k, discrete_cdf(k, k, pk)); figure; plot(k, 1-discrete_cdf(k, k, pk)); L = 1000; k = [1:1:L]; cL = sum(1./k); pk = (1/cL)./k; figure; plot(k, discrete_pdf(k, k, pk)); figure; plot(k, discrete_cdf(k, k, pk)); figure; plot(k, 1-discrete_cdf(k, k, pk)); (b) m = 20; k = [1:1:m]; pk = (1/2).^k; figure; semilogy(k, pk); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3.80 3-52 The following Octave commands will plot the Lorenze curves: L = 10; k = [1:1:L]; cL = sum(1./k); pk = (1/cL)./k; Wk = k./L; Fk = discrete_cdf(k, k, pk); figure; plot(Fk, Wk); L = 100; k = [1:1:L]; cL = sum(1./k); pk = (1/cL)./k; Wk = k./L; Fk = discrete_cdf(k, k, pk); figure; plot(Fk, Wk); L = 1000; k = [1:1:L]; cL = sum(1./k); pk = (1/cL)./k; Wk = k./L; Fk = discrete_cdf(k, k, pk); figure; plot(Fk, Wk); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3.81 3-53 The following Octave commands will plot the requested curves: figure; hold on; n = 100; p = 0.1; k = [0:1:n]; Wk = zeros(1, n+1); for i = 0:n, v = [0:i]; Wk(i+1) = sum(v.*binomial_pdf(v, n, p))./(n*p); end; Fk = binomial_cdf(k, n, p); plot(Fk, Wk); n = 100; p = 0.5; k = [0:1:n]; Wk = zeros(1, n+1); for i = 0:n, v = [0:i]; Wk(i+1) = sum(v.*binomial_pdf(v, n, p))./(n*p); end; Fk = binomial_cdf(k, n, p); plot(Fk, Wk); n = 100; p = 0.9; k = [0:1:n]; Wk = zeros(1, n+1); for i = 0:n, v = [0:i]; Wk(i+1) = sum(v.*binomial_pdf(v, n, p))./(n*p); end; Fk = binomial_cdf(k, n, p); plot(Fk, Wk); 3.82 (a) (b) (c) © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-54 3.83 The following Octave commands will generate the requested samples of the Zipf random variable and the requested plots. L = 10; k = [1:1:L]; cL = sum(1./k); pk = (1/cL)./k; Sk = discrete_rnd(200, k, pk); figure; plot(Sk); figure; hist(Sk, k); L = 100; k = [1:1:L]; cL = sum(1./k); pk = (1/cL)./k; Sk = discrete_rnd(200, k, pk); figure; plot(Sk); figure; hist(Sk, k); L = 1000; k = [1:1:L]; cL = sum(1./k); pk = (1/cL)./k; Sk = discrete_rnd(200, k, pk); figure; plot(Sk); figure; hist(Sk, k); 3.84 The following Octave commands generate the samples of the St. Peter’s Paradox random variable and the requested plots. m = 20; k = [1:1:m]; pk = (1/2).^k; Sk = discrete_rnd(200, k, pk); figure; plot(Sk); figure; hist(Sk, k); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3.85 3-55 The following Octave commands generate the requested pairs and plots: (a) k = [1:10]; pk = ones(1,10)./10; Sx = discrete_rnd(200, k, pk); Sy = discrete_rnd(200, k, pk); figure; hist(Sx, k); figure; hist(Sy, k); (b) Sz = Sx + Sy; figure; hist(Sz, [2:20]); (c) Sw = Sx .* Sy; figure; hist(Sw, 10); (d) Sv = Sx ./ Sy; figure; hist(Sv, 10); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3.86 3-56 The following Octave commands generate the requested pairs and plots: (a) Sx = binomial_rnd(8, 0.5, 1, 200); Sy = binomial_rnd(4, 0.5, 1, 200); figure; hist(Sx, [0:8]); figure; hist(Sy, [0:4]); (b) Sz = Sx + Sy; figure; hist(Sz, [0:12]); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3.87 3-57 The following Octave commands generate the requested pairs and plots: (a) Sx = poisson_rnd(5, 1, 200); Sy = poisson_rnd(10, 1, 200); figure; hist(Sx, [0:15]); figure; hist(Sy, [0:20]); (b) Sz = Sx + Sy; figure; hist(Sz, [0:35]); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-58 Problems Requiring Cumulative Knowledge 3.88 3.89 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-59 3.90 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-60 3.91 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 3-61 3.92 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-1 Chapter 4: One Random Variable 4.1 The Cumulative Distribution Function 4.1 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-2 4.2 4.3 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-2 4.4 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-4 4.5 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-5 4.6 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-6 4.7 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-7 4.8 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-8 4.9 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-9 4.10 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-10 4.11 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-11 4.12 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-12 4.13 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-13 4.14 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-14 4.15 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-15 4.16 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4.2 4-16 The Probability Density Function 4.17 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-17 4.18 4.19 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-18 4.20 4.21 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-19 4.22 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-20 4.23 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-21 4.24 4.25 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-22 4.26 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-23 4.27 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-24 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-25 4.28 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-26 4.29 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-27 4.30 4.31 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-28 4.32 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-29 4.33 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-30 4.34 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. 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For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-43 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-44 4.55 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. 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For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-85 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-86 4.99 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-87 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-88 4.100 4.101 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4.7 4-89 Transform Methods 4.102 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-90 4.103 4.104 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-91 4.105 4.106 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-92 4.107 4.108 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-93 4.109 4.110 4.111 4.112 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-94 4.113 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-95 4.114 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-96 4.115 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-97 4.116 4.117 4.118 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-98 4.119 4.120 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-99 4.121 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4.8 4-100 Basic Reliability Calculations 4.122 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-101 4.123 4.124 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-102 4.125 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-103 4.126 4.127 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-104 4.128 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-105 4.129 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-106 4.130 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-107 4.131 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-108 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-109 4.132 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-110 4.133 (a) © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4.9 4.134 4-111 Computer Methods for Generating Random Variables The following Octave commands generate the requested plots: (a) x = [-4:0.01:4]; y0 = normal_pdf(x, -2, 1); y1 = normal_pdf(x, 2, 1); figure; hold on; plot(x, y0, "1"); plot(x, y1, "3"); (b) x = [-5:0.01:5]; y0 = 1-normal_cdf(x, -2, 1); y1 = 1-normal_cdf(x, 2, 1); ey0 = e.^(-(x+2).^2/2); ey1 = e.^(-(x-2).^2/2); figure; hold on; plot(x, y0, "1"); plot(x, y1, "3"); plot(x, ey0, "1"); plot(x, ey1, "3"); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4.135 4-112 The following Octave commands generate plots of the pdf and cdf of the gamma random variable: (a) x = [0:0.01:15]; figure; hold on; plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, figure; hold on; plot(x, gamma_cdf(x, plot(x, gamma_cdf(x, plot(x, gamma_cdf(x, 1, 1), "1"); 2, 1), "2"); 4, 1), "3"); 1, 1), "1"); 2, 1), "2"); 4, 1), "3"); (b) x = [0:0.01:15]; figure; hold on; plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, figure; hold on; plot(x, gamma_cdf(x, plot(x, gamma_cdf(x, plot(x, gamma_cdf(x, plot(x, gamma_cdf(x, 1/2, 1/2), "1"); 1, 1/2), "2"); 3/2, 1/2), "3"); 5/2, 1/2), "4"); 1/2, 1/2), "1"); 1, 1/2), "2"); 3/2, 1/2), "3"); 5/2, 1/2), "4"); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-113 4.136 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4.137 4-114 The following Octave commands generate the requested samples and plots: (a) x = [-6:0.01:6]; u = rand(1, 1000); %Multiply all values by discretely generated -1 or 1 z = -log(u).*discrete_rnd(length(u), [-1 1], [0.5 0.5]); figure; hold on; %Normalize to 2 because bar width is 0.5 hist(z, [-6:0.5:6], 2); plot(x, laplace_pdf(x), "1"); (b) x = [1:0.01:10]; u = rand(1, 1000); k = 1.5; z = u.^(-1/k); figure; hold on; hist(z, [1.25:0.5:10], 2); plot(x, k./x.^(k+1)); x = [1:0.01:10]; u = rand(1, 1000); k = 2; z = u.^(-1/k); figure; hold on; hist(z, [1.25:0.5:10], 2); plot(x, k./x.^(k+1)); x = [1:0.01:10]; u = rand(1, 1000); k = 2.5; z = u.^(-1/k); figure; hold on; hist(z, [1.25:0.5:10], 2); plot(x, k./x.^(k+1)); (c) x = [0:0.01:5]; u = rand(1, 1000); b = 0.5; z = log(1./u).^(1./b); figure; hold on; hist(z, [0:0.25:5], 4); plot(x, b.*x.^(b-1).*e.^(-x.^b), "1"); x = [0:0.01:5]; u = rand(1, 1000); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-115 b = 2; z = log(1./u).^(1./b); figure; hold on; hist(z, [0:0.125:5], 8); plot(x, b.*x.^(b-1).*e.^(-x.^b), "1"); x = [0:0.01:5]; u = rand(1, 1000); b = 3; z = log(1./u).^(1./b); figure; hold on; hist(z, [0:0.125:5], 8); plot(x, b.*x.^(b-1).*e.^(-x.^b), "1"); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-116 4.138 4.139 4.140 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4-117 4.141 The following Octave commands describe the function for performing the rejection method and the code to call the function: function z = gaussian_rejection_method(N) z = zeros(1, N); k = 1; while k 20) i = 20; end if (j>15) j = 20; end Freq(i,j)=Freq(i,j)+1; end chi = 0; for i=1:1:19 for j=1:1:19 expec(i,j)=expected1(i)*expected2(j); chi=chi+(Freq(i,j)/maxrep-expec(i,j))^2/(expec(i,j)); end end chi figure(10); subplot(2,1,1); mesh(expec); ylabel('theoretical prob'); subplot(2,1,2); mesh(Freq/maxrep); ylabel('simulation'); 9-138 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-139 9.129 (a) %lambda*t=n*p, if lambda=1 then n*p should be t clear all; close all; n=2000; p=200/n; maxrep = 1; N(1:maxrep,1:1:n)=0; A(1:maxrep,1:1:n)=0; D(1:maxrep,1:1:n)=0; L(1:maxrep,1:1:n)=0; % # of customers leaving at time t in (1,n) for r=1:1:maxrep %assuming we will have maximum of 3000 customer and their service time distribution is exponential. T(1:1:3000) = floor(exponential_inv(rand(1,3000), 5*n/200)); N(r,1)=0; A(r,1)=0; D(r,1)=0; cust=1; for i=2:1:n if (rand < p) A(r,i)=A(r,i-1)+1; N(r,i)=N(r,i-1)+1; if (i+T(cust) < n) L(r,i+T(cust))=L(r,i+T(cust))+1; end cust = cust+1; else A(r,i)=A(r,i-1); N(r,i)=N(r,i-1); end D(r,i)=D(r,i-1)+L(r,i); N(r,i)=N(r,i)-L(r,i); end end plot(1:1:n,A(1,:),1:1:n,N(1,:),1:1:n,D(1,:)); legend('Arrival','Number in system','Departure'); (b) (c) First part of code is the same as part a. %lambda*t=n*p, if lambda=1 then n*p should be 200 clear all; close all; n=2000; p=200/n; t=n/200; % one second is t steps A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-140 maxrep = 100; N(1:maxrep,1:1:n)=0; A(1:maxrep,1:1:n)=0; D(1:maxrep,1:1:n)=0; L(1:maxrep,1:1:n)=0; % # of customers leaving at time t in (1,n) for r=1:1:maxrep %assuming we will have maximum of 3000 customer and their service time distribution is exponential. T(1:1:3000) = floor(exponential_inv(rand(1,3000), 5*t)); N(r,1)=0; A(r,1)=0; D(r,1)=0; cust=1; for i=2:1:n if (rand < p) A(r,i)=A(r,i-1)+1; N(r,i)=N(r,i-1)+1; if (i+T(cust) < n) L(r,i+T(cust))=L(r,i+T(cust))+1; end cust = cust+1; else A(r,i)=A(r,i-1); N(r,i)=N(r,i-1); end D(r,i)=D(r,i-1)+L(r,i); N(r,i)=N(r,i)-L(r,i); end end figure(1); plot(1:1:n,A(1,:),1:1:n,N(1,:),1:1:n,D(1,:)); legend('Arrival','Number in system','Departure'); d50=hist(D(:,50*t),1:max(D(:,50*t)))/maxrep; d100=hist(D(:,100*t),1:max(D(:,100*t)))/maxrep; d150=hist(D(:,150*t),1:max(D(:,150*t)))/maxrep; d200=hist(D(:,200*t),1:max(D(:,200*t)))/maxrep; %we know that theoretical pmf for D(t) is poisson random variables %pmf with rate lambda*(t-5) expected50=poisson_pdf(1:max(D(:,50*t)), 45); expected100=poisson_pdf(1:max(D(:,100*t)), 95); expected150=poisson_pdf(1:max(D(:,150*t)), 145); expected200=poisson_pdf(1:max(D(:,200*t)), 195); figure(2); plot(1:max(D(:,50*t)),d50, '-*', 1:max(D(:,50*t)),expected50, '-o'); legend('histogram','poisson'); title('D(50)'); figure(3); plot(1:max(D(:,100*t)),d100, '-*', 1:max(D(:,100*t)),expected100, '-o'); legend('histogram','poisson'); title('D(100)'); figure(4); A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-141 plot(1:max(D(:,150*t)),d150, '-*', 1:max(D(:,150*t)),expected150, '-o'); legend('histogram','poisson'); title('D(150)'); figure(5); plot(1:max(D(:,200*t)),d200, '-*', 1:max(D(:,200*t)),expected200, '-o'); legend('histogram','poisson'); title('D(200)'); %chi-square goodness-of-fit test: sum(y-expected)^2/expected p50=sum((d50-expected50).^2./expected50) p100=sum((d100-expected100).^2./expected100) p150=sum((d150-expected150).^2./expected150) p200=sum((d200-expected200).^2./expected200) (d) %lambda*t=n*p, if lambda=1 then n*p should be 200 seconds clear all; close all; n=2000; p=200/n; t=n/200; % one second is t steps maxrep = 100; N(1:maxrep,1:1:n)=0; A(1:maxrep,1:1:n)=0; D(1:maxrep,1:1:n)=0; L(1:maxrep,1:1:n)=0; % # of customers leaving at time t in (1,n) for r=1:1:maxrep N(r,1)=0; A(r,1)=0; D(r,1)=0; cust=1; for i=2:1:n if (rand < p) A(r,i)=A(r,i-1)+1; N(r,i)=N(r,i-1)+1; if (i+5*t < n) L(r,i+5*t)=L(r,i+5*t)+1; end cust = cust+1; else A(r,i)=A(r,i-1); N(r,i)=N(r,i-1); end D(r,i)=D(r,i-1)+L(r,i); N(r,i)=N(r,i)-L(r,i); end end figure(1); plot(1:1:n,A(1,:),1:1:n,N(1,:),1:1:n,D(1,:)); legend('Arrival','Number in system','Departure'); d50=hist(D(:,50*t),1:max(D(:,50*t)))/maxrep; A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-142 d100=hist(D(:,100*t),1:max(D(:,100*t)))/maxrep; d150=hist(D(:,150*t),1:max(D(:,150*t)))/maxrep; d200=hist(D(:,200*t),1:max(D(:,200*t)))/maxrep; %we know that theoretical pmf for D(t) is poisson random variables %pmf with rate lambda*(t-5) expected50=poisson_pdf(1:max(D(:,50*t)), 45); expected100=poisson_pdf(1:max(D(:,100*t)), 95); expected150=poisson_pdf(1:max(D(:,150*t)), 145); expected200=poisson_pdf(1:max(D(:,200*t)), 195); figure(2); plot(1:max(D(:,50*t)),d50, '-*', 1:max(D(:,50*t)),expected50, '-o'); legend('histogram','poisson'); title('D(50)'); figure(3); plot(1:max(D(:,100*t)),d100, '-*', 1:max(D(:,100*t)),expected100, '-o'); legend('histogram','poisson'); title('D(100)'); figure(4); plot(1:max(D(:,150*t)),d150, '-*', 1:max(D(:,150*t)),expected150, '-o'); legend('histogram','poisson'); title('D(150)'); figure(5); plot(1:max(D(:,200*t)),d200, '-*', 1:max(D(:,200*t)),expected200, '-o'); legend('histogram','poisson'); title('D(200)'); %chi-square goodness-of-fit test: sum(y-expected)^2/expected p50=sum((d50-expected50).^2./expected50) p100=sum((d100-expected100).^2./expected100) p150=sum((d150-expected150).^2./expected150) p200=sum((d200-expected200).^2./expected200) A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.130 (a) clear all; close all; s=100; %each second divided to 100 steps %alpha=1, so h=sqrt(alpha/s)=0.1 alpha=1; h=sqrt(alpha/s); n=s*3.5; maxrep=1000; x(1:maxrep,1:n)=0; inc(1:maxrep,3)=0; for r=1:1:maxrep x(r,1)=0; for i=2:1:n if (rand < 0.5) x(r,i)=x(r,i-1)+h; else x(r,i)=x(r,i-1)-h; end end inc(r,1)=x(r,0.5*s)-x(r,1); inc(r,2)=x(r,1.5*s)-x(r,0.5*s+1); inc(r,3)=x(r,3.5*s)-x(r,1.5*s+1); end [y b]=hist(inc(:,1),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(0.5)); sno=sum(no); figure(1); plot(-6:0.4:6,no/sno,b,y/maxrep) legend('normal pdf','simulation'); title('inc (0,0.5)'); [y b]=hist(inc(:,2),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(1.5-0.5)); sno=sum(no); figure(2); plot(-6:0.4:6,no/sno,b,y/maxrep) legend('normal pdf','simulation'); title('inc (0.5,1.5)'); [y b]=hist(inc(:,3),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-8:0.4:8,0,sqrt(3.5-1.5)); sno=sum(no); figure(3); plot(-8:0.4:8,no/sno,b,y/maxrep) legend('normal pdf','simulation'); title('inc (3.5,1.5)'); 9-143 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering (b) clear all; close all; s=100; %each second divided to 100 steps %alpha=1, so h=sqrt(alpha/s)=0.1 alpha=1; h=sqrt(alpha/s); n=s*3.5; maxrep=100; x(1:maxrep,1:n)=0; inc(1:maxrep,3)=0; for r=1:1:maxrep x(r,1)=0; for i=2:1:n if (rand < 0.5) x(r,i)=x(r,i-1)+h; else x(r,i)=x(r,i-1)-h; end end inc(r,1)=x(r,0.5*s)-x(r,1); inc(r,2)=x(r,1.5*s)-x(r,0.5*s+1); inc(r,3)=x(r,3.5*s)-x(r,1.5*s+1); end [y b]=hist(inc(:,1),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(0.5)); sno=sum(no); figure(1); plot(-6:0.4:6,no/sno, '--',b,y/maxrep,'-o') legend('normal','histogram'); title('inc(0,0.5)'); pdf1=no/sno; [y b]=hist(inc(:,2),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(1.5-0.5)); sno=sum(no); figure(2); plot(-6:0.4:6,no/sno, '--',b,y/maxrep,'-o') legend('normal','histogram'); title('inc(0.5,1.5)'); pdf2=no/sno; [y b]=hist(inc(:,3),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(3.5-1.5)); sno=sum(no); pdf3=no/sno; figure(3); plot(-6:0.4:6,no/sno, '--',b,y/maxrep,'-o') legend('normal','histogram'); title('inc(1.5,3.5)'); 9-144 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering value=-6:0.4:6; vallen=length(value); %number of values is 12*2.5=30 chi=0; p0(1:1:vallen,1:1:vallen)=0; freq(1:1:vallen,1:1:vallen)=0; for i=2:1:vallen-1 for j=2:1:vallen-1 p0(i,j)=pdf1(i)*pdf2(j); for r=1:1:maxrep if (inc(r,1) ((value(i-1)+value(i))/2)) if (inc(r,2) ((value(j-1)+value(j))/2)) freq(i,j) = freq(i,j)+1; end end end freq(i,j)=freq(i,j)/maxrep; chi=chi+((freq(i,j)-p0(i,j))^2/p0(i,j)); end end chi %(chi for 10000 sim is 0.02), (chi for 1000 sim is 0.35) figure(4); subplot(2,1,1); mesh(value,value,p0); axis([-5 5 -5 5]); zlabel('theoretical prob'); subplot(2,1,2); mesh(value,value,freq); axis([-5 5 -5 5]); zlabel('simulation'); title('Dependency check, inc (0,0.5) & inc(0.5,1.5)'); 9-145 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.131 (a) clear all; close all; s=100; %each second divided to 100 steps %each sample is a gaussian random variable with zero mean, and variance %alpha*t. Since alpha=1, then just generate randn(0,step)+prevvalue n=3.5*s; alpha=1; maxrep=100; x(1:maxrep,1:n)=0; inc(1:maxrep,3)=0; for r=1:1:maxrep x(r,1)=0; for i=2:1:n x(r,i)=x(r,i-1)+sqrt(1/s)*randn; end inc(r,1)=x(r,0.5*s)-x(r,1); inc(r,2)=x(r,1.5*s)-x(r,0.5*s+1); inc(r,3)=x(r,3.5*s)-x(r,1.5*s+1); end [y b]=hist(inc(:,1),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(0.5)); sno=sum(no); figure(1); plot(-6:0.4:6,no/sno,b,y/maxrep) legend('normal pdf','simulation'); [y b]=hist(inc(:,2),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(1.5-0.5)); sno=sum(no); figure(2); plot(-6:0.4:6,no/sno,b,y/maxrep) legend('normal pdf','simulation'); [y b]=hist(inc(:,3),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-8:0.4:8,0,sqrt(3.5-1.5)); sno=sum(no); figure(3); plot(-8:0.4:8,no/sno,b,y/maxrep) legend('normal pdf','simulation'); (b) clear all; close all; s=100; %each second divided to 100 steps %each sample is a gaussian random variable with zero mean, and variance %alpha*t. Since alpha=1, then just generate randn(0,step)+prevvalue 9-146 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering n=3.5*s; alpha=1; maxrep=100; x(1:maxrep,1:n)=0; inc(1:maxrep,3)=0; for r=1:1:maxrep x(r,1)=0; for i=2:1:n x(r,i)=x(r,i-1)+sqrt(1/s)*randn; end inc(r,1)=x(r,0.5*s)-x(r,1); inc(r,2)=x(r,1.5*s)-x(r,0.5*s+1); inc(r,3)=x(r,3.5*s)-x(r,1.5*s+1); end [y b]=hist(inc(:,1),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(0.5)); sno=sum(no); figure(1); plot(-6:0.4:6,no/sno, '--',b,y/maxrep,'-o') legend('normal','histogram'); title('inc(0,0.5)'); pdf1=no/sno; [y b]=hist(inc(:,2),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(1.5-0.5)); sno=sum(no); figure(2); plot(-6:0.4:6,no/sno, '--',b,y/maxrep,'-o') legend('normal','histogram'); title('inc(0.5,1.5)'); pdf2=no/sno; [y b]=hist(inc(:,3),-6:0.4:6); [sum(y/maxrep) mean(y/maxrep) var(y/maxrep)] no=normal_pdf(-6:0.4:6,0,sqrt(3.5-1.5)); sno=sum(no); pdf3=no/sno; figure(3); plot(-6:0.4:6,no/sno, '--',b,y/maxrep,'-o') legend('normal','histogram'); title('inc(1.5,3.5)'); value=-6:0.4:6; vallen=length(value); %number of values is 12*2.5=30 chi=0; p0(1:1:vallen,1:1:vallen)=0; freq(1:1:vallen,1:1:vallen)=0; for i=2:1:vallen-1 for j=2:1:vallen-1 p0(i,j)=pdf1(i)*pdf2(j); for r=1:1:maxrep 9-147 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering if (inc(r,1) ((value(i-1)+value(i))/2)) if (inc(r,2) ((value(j-1)+value(j))/2)) freq(i,j) = freq(i,j)+1; end end end freq(i,j)=freq(i,j)/maxrep; chi=chi+((freq(i,j)-p0(i,j))^2/p0(i,j)); end end chi %(chi for 10000 sim is 0.02), (chi for 1000 sim is 0.35) figure(4); subplot(2,1,1); mesh(value,value,p0); axis([-5 5 -5 5]); zlabel('theoretical prob'); subplot(2,1,2); mesh(value,value,freq); axis([-5 5 -5 5]); zlabel('simulation'); title('Dependency check, inc (0,0.5) & inc(0.5,1.5)'); 9-148 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering Problems Requiring Cumulative Knowledge 9.132 4.1 9-149 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-150 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.133 a) 9-151 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.134 9-152 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-153 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.135 9-154 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-155 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.136 9-156 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.137 9-157 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.138 9-158 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.139 9.140 9-159 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.141 9-160 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering Chapter 9: Random Processes – Part I 9.1 & 9.2 9.1 Definition and Specification of a Stochastic Process 9-1 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.2 9-2 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.3 9-3 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.4 9-4 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-5 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.5 9-6 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.6 9-7 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-8 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-9 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.7 9-10 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-11 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.8 9-12 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-13 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-14 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.9 9.10 9-15 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.11 9-16 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.12 9-17 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.13 9-18 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.14 9-19 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.15 9-20 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-21 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.16 9-22 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-23 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.17 9-24 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.18 9-25 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.19 9-26 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-27 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.20 9-28 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.3 9.21 Sum Process, Binomial Counting Process, and Random Walk 9-29 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.22 9.23 9-30 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.24 9-31 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-32 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-33 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.25 9-34 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.26 9.27 9-35 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.28 9.29 9.30 9-36 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.31 9-37 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.32 9-38 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9-39 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.33 9-40 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.4 9.34 9.35 9.36 Poisson and Associated Random Processes 9-41 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.37 9-42 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.38 9-43 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.39 9-44 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.40 9-45 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.41 9.42 9-46 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.43 9-47 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.44 9-48 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.45 9.46 9-49 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.47 9-50 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.48 9.49 9-51 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.5 Gaussian Random Processes, Wiener Process and Brownian Motion 9.50 9-52 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.51 9.52 9-53 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.53 9-54 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.54 9.55 9-55 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.56 9.57 9-56 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.58 9-57 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.59 9-58 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 9.60 End of Section: END OF PART I. 9-59 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10-1 Chapter 10: Analysis and Processing of Random Signals 10.1 Power Spectral Density 10.1 10.2 10.3 where A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.4 P10.2 and g(x) as in Fig. P10.1. 10-2 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.5 10-3 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.6 10.4 10.7 10.8 10-4 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.9 10.10 10-5 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.11 10-6 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.12 10.13 10-7 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.14 10-8 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.15 Xn = cos(2π f0n + Θ) SX(f) 10-9 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.16 10.17 10-10 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.18 10.19 10-11 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.20 10-12 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.21 %P.10.21 clear all; close all; samples = 50; len=128; x=randn(len,samples); xf=fft(x); pf=sqrt(xf'.*conj(xf'))/len; sf=mean(pf); sf1=mean(pf(1:10,:)); sf2=mean(pf(1:20,:)); sf3=mean(pf(1:30,:)); sf4=mean(pf(1:40,:)); subplot(2,3,1); plot(1:len,sf); axis([1 len 0.06 0.11]); title('All (50) realizations'); subplot(2,3,2); plot(1:len,sf1); axis([1 len 0.06 0.11]); title('10 realizations'); subplot(2,3,3); plot(1:len,sf2); axis([1 len 0.06 0.11]); title('20 realizations'); subplot(2,3,4); plot(1:len,sf3); axis([1 len 0.06 0.11]); title('30 realizations'); subplot(2,3,5); plot(1:len,sf4); axis([1 len 0.06 0.11]); title('40 realizations'); 10-13 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.2 Response of Linear Systems to Random Signals 10.22 10.23 10-14 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.24 10.25 10-15 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.26 10-16 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.27 10.28 10-17 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10-18 10.29 10.42 10.30 10.45a: A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.31 10-19 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.32 10-20 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.33 10-21 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.34 10-22 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.35 10.33 10-23 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.36 10-24 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.37 10.38 10-25 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.39 10-26 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.40 10.13 10-27 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.41 10-28 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.42 10-29 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.43 10.42, 10-30 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.44 10-31 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.3 Bandlimited Random Processes 10.45 10-32 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10-33 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10-34 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.46 10-35 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10-36 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.47 10-37 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10-38 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.48 10-39 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10-40 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.49 10.4 10-41 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.50 10.51 10-42 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.52 10-43 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.53 10-44 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.54 10-45 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.55 Eqn. 10.67b implies 10.7 10-46 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.4 Optimum Linear Systems 10.56 10-47 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.57 Eqn. 10.83 implies 10-48 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.58 Eqn. 10.83 is then 10-49 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.59 10-50 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.60 10-51 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.61 10-52 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.62 10.63 10-53 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.64 10.30 10.92 10.65 10-54 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.66 10.67 10-55 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.68 10.89 10.14 10.64 10-56 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10-57 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.69 10.99 10-58 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.70 10.30 10.25 10-59 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.71 10-60 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.72 10-61 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10-62 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering *10.5 The Kalman Filter 10.73 10.74 10-63 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.75 10.76 10-64 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering *10.6 Estimating the Power Spectral Density 10.77 10-65 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.78 10.79 10-66 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.80 10-67 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.7 Numerical Techniques for Processing Random Signals 10.81 10-68 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering (d) The Octave code for part d: %P10.81 %part d clear all; close all; N=1024; M=N/2; W0=10; t0=1/(4.6*W0); f0=4.6*W0/N; T=N/(9.2*W0); m=-M:1:M-1; Sx=exp(-1*(m.^2*f0^2)/(2*W0^2)); k=-M:1:M-1; Rfex=fft(Sx); Rf=Rfex.*conj(Rfex)/N; Rex=Sx*exp(-j*2*pi*m'*k/N); Rex2=f0*sqrt(Rex.*conj(Rex)); %Rx=sum(Sx*exp(-j*2*pi*m*k'/N)); Rx=exp(-1*(m.^2*t0^2)*(2*pi^2*W0^2))*(sqrt(2*pi)*W0); subplot(2,1,1) plot(m,Rx,m,Rex2); legend('Actual','FFT'); subplot(2,1,2) plot(Rx,Rex2); title('Actual vs. FFT'); 10-69 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.82 %P10.82 %Part a clear all; close all; N=256; M=N/2; k=-M:1:M-1; alpha=0.75; %part a Rx=4*alpha.^abs(k); Sx=fft(Rx); figure(1); plot((k+M)/N,sqrt(Sx.*conj(Sx))); title('P10.82a'); %part b Rx=4*0.5.^abs(k)+16*0.25.^abs(k); Sx=fft(Rx); figure(2); plot((k+M)/N,sqrt(Sx.*conj(Sx))); title('P10.82b'); %part c Rx=0.5*cos(2*pi*0.1*k); Sx=fft(Rx); figure(3); plot((k+M)/N,sqrt(Sx.*conj(Sx))); title('P10.82c'); 10-70 A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 10.83 %P10.83 %Part a clear all; close all; N=256; M=N/2; k=-M:1:M-1; fc=1/8; Sx(1:1:N)= 1; for i=1:1:N if (((i-M-1)/N) >= -fc) if (((i-M-1)/N) = -fc) if (((i-M-1)/N) |
Probability statistics and random processes for electrical engineering solution
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