chapter 5: pairs of random variables - 기록용...

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-1 Probability, Statistics, and Random Processes for Electrical Engineering

Chapter 5: Pairs of Random Variables 5.1 Two Random Variables

5.1

© 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 5-2 Probability, Statistics, and Random Processes for Electrical Engineering

5.2

© 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 5-3 Probability, Statistics, and Random Processes for Electrical Engineering

5.3

© 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 5-4 Probability, Statistics, and Random Processes for Electrical Engineering

5.4

© 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 5-5 Probability, Statistics, and Random Processes for Electrical Engineering

5.5

© 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 5-6 Probability, Statistics, and Random Processes for Electrical Engineering

5.6

5.7

© 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 5-7 Probability, Statistics, and Random Processes for Electrical Engineering

5.8

© 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 5-8 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-9 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-10 Probability, Statistics, and Random Processes for Electrical Engineering

5.2 Pairs of Discrete Random Variables

5.9

© 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 5-11 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-12 Probability, Statistics, and Random Processes for Electrical Engineering

5.10

© 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 5-13 Probability, Statistics, and Random Processes for Electrical Engineering

5.11

© 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 5-14 Probability, Statistics, and Random Processes for Electrical Engineering

5.12

© 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 5-15 Probability, Statistics, and Random Processes for Electrical Engineering

5.13

© 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 5-16 Probability, Statistics, and Random Processes for Electrical Engineering

5.14

© 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 5-17 Probability, Statistics, and Random Processes for Electrical Engineering

5.15

© 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 5-18 Probability, Statistics, and Random Processes for Electrical Engineering

5.3 The Joint pdf of X and Y

5.16

© 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 5-19 Probability, Statistics, and Random Processes for Electrical Engineering

5.17

© 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 5-20 Probability, Statistics, and Random Processes for Electrical Engineering

5.17

© 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 5-21 Probability, Statistics, and Random Processes for Electrical Engineering

5.18

© 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 5-22 Probability, Statistics, and Random Processes for Electrical Engineering

5.19

© 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 5-23 Probability, Statistics, and Random Processes for Electrical Engineering

5.19

© 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 5-24 Probability, Statistics, and Random Processes for Electrical Engineering

5.20

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-25 Probability, Statistics, and Random Processes for Electrical Engineering

5.21

© 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 5-26 Probability, Statistics, and Random Processes for Electrical Engineering

5.22

© 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 5-27 Probability, Statistics, and Random Processes for Electrical Engineering

5.23

© 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 5-28 Probability, Statistics, and Random Processes for Electrical Engineering

5.24

© 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 5-29 Probability, Statistics, and Random Processes for Electrical Engineering

5.4 The Joint cdf of Two Continuous Random Variables

5.25

© 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 5-30 Probability, Statistics, and Random Processes for Electrical Engineering

5.26

© 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 5-31 Probability, Statistics, and Random Processes for Electrical Engineering

5.27

© 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 5-32 Probability, Statistics, and Random Processes for Electrical Engineering

5.28

5.28c

© 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 5-33 Probability, Statistics, and Random Processes for Electrical Engineering

5.29

5.30

© 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 5-34 Probability, Statistics, and Random Processes for Electrical Engineering

5.31

© 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 5-35 Probability, Statistics, and Random Processes for Electrical Engineering

5.31

5.32

5.33

© 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 5-36 Probability, Statistics, and Random Processes for Electrical Engineering

5.34

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-37 Probability, Statistics, and Random Processes for Electrical Engineering

5.35

© 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 5-38 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-39 Probability, Statistics, and Random Processes for Electrical Engineering

5.36

© 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 5-40 Probability, Statistics, and Random Processes for Electrical Engineering

5.36b

5.36c

© 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 5-41 Probability, Statistics, and Random Processes for Electrical Engineering

5.36c

© 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 5-42 Probability, Statistics, and Random Processes for Electrical Engineering

5.5 Independence of Two Random Variables

5.37

5.38

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-43 Probability, Statistics, and Random Processes for Electrical Engineering

5.39

5.40

© 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 5-44 Probability, Statistics, and Random Processes for Electrical Engineering

5.41

© 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 5-45 Probability, Statistics, and Random Processes for Electrical Engineering

5.42

5.43

5.44

5.45

© 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 5-46 Probability, Statistics, and Random Processes for Electrical Engineering

5.46

5.47

© 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 5-47 Probability, Statistics, and Random Processes for Electrical Engineering

5.48

5.49

© 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 5-48 Probability, Statistics, and Random Processes for Electrical Engineering

5.50

© 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 5-49 Probability, Statistics, and Random Processes for Electrical Engineering

5.51

© 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 5-50 Probability, Statistics, and Random Processes for Electrical Engineering

5.52

© 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 5-51 Probability, Statistics, and Random Processes for Electrical Engineering

5.53

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-52 Probability, Statistics, and Random Processes for Electrical Engineering

5.54

5.55

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-53 Probability, Statistics, and Random Processes for Electrical Engineering

5.6 Joint Moments and Expected Value of a Function of Two Random Variables

5.56

5.57

5.58

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-54 Probability, Statistics, and Random Processes for Electrical Engineering

5.60

5.59

© 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 5-55 Probability, Statistics, and Random Processes for Electrical Engineering

5.61

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-56 Probability, Statistics, and Random Processes for Electrical Engineering

5.62

© 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 5-57 Probability, Statistics, and Random Processes for Electrical Engineering

5.63

© 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 5-58 Probability, Statistics, and Random Processes for Electrical Engineering

5.64

© 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 5-59 Probability, Statistics, and Random Processes for Electrical Engineering

5.65

© 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 5-60 Probability, Statistics, and Random Processes for Electrical Engineering

5.66

© 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 5-61 Probability, Statistics, and Random Processes for Electrical Engineering

5.67

5.68

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-62 Probability, Statistics, and Random Processes for Electrical Engineering

5.69

© 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 5-63 Probability, Statistics, and Random Processes for Electrical Engineering

5.70

© 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 5-64 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-65 Probability, Statistics, and Random Processes for Electrical Engineering

5.71

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-66 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-67 Probability, Statistics, and Random Processes for Electrical Engineering

5.72

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-68 Probability, Statistics, and Random Processes for Electrical Engineering

5.73

5.74

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-69 Probability, Statistics, and Random Processes for Electrical Engineering

5.7 Conditional Probability and Conditional Expectation

9

5.75

© 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 5-70 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-71 Probability, Statistics, and Random Processes for Electrical Engineering

5.76

© 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 5-72 Probability, Statistics, and Random Processes for Electrical Engineering

5.77

© 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 5-73 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-74 Probability, Statistics, and Random Processes for Electrical Engineering

5.78

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-75 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-76 Probability, Statistics, and Random Processes for Electrical Engineering

5.79

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-77 Probability, Statistics, and Random Processes for Electrical Engineering

5.80

© 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 5-78 Probability, Statistics, and Random Processes for Electrical Engineering

5.81

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-79 Probability, Statistics, and Random Processes for Electrical Engineering

5.82

© 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 5-80 Probability, Statistics, and Random Processes for Electrical Engineering

5.83

© 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 5-81 Probability, Statistics, and Random Processes for Electrical Engineering

5.84

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-82 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-83 Probability, Statistics, and Random Processes for Electrical Engineering

5.85

© 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 5-84 Probability, Statistics, and Random Processes for Electrical Engineering

5.86

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-85 Probability, Statistics, and Random Processes for Electrical Engineering

5.87

© 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 5-86 Probability, Statistics, and Random Processes for Electrical Engineering

5.88

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-87 Probability, Statistics, and Random Processes for Electrical Engineering

5.89

© 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 5-88 Probability, Statistics, and Random Processes for Electrical Engineering

5.8 Functions of Two Random Variables

5.90

5.91

© 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 5-89 Probability, Statistics, and Random Processes for Electrical Engineering

5.92

© 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 5-90 Probability, Statistics, and Random Processes for Electrical Engineering

5.93

© 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 5-91 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-92 Probability, Statistics, and Random Processes for Electrical Engineering

5.95

5.94

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-93 Probability, Statistics, and Random Processes for Electrical Engineering

5.96

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-94 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-95 Probability, Statistics, and Random Processes for Electrical Engineering

(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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-96 Probability, Statistics, and Random Processes for Electrical Engineering

5.97

© 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 5-97 Probability, Statistics, and Random Processes for Electrical Engineering

5.98

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-98 Probability, Statistics, and Random Processes for Electrical Engineering

5.99

© 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 5-99 Probability, Statistics, and Random Processes for Electrical Engineering

5.100

© 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 5-100 Probability, Statistics, and Random Processes for Electrical Engineering

5.101

© 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 5-101 Probability, Statistics, and Random Processes for Electrical Engineering

5.103

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-102 Probability, Statistics, and Random Processes for Electrical Engineering

5.104

© 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 5-103 Probability, Statistics, and Random Processes for Electrical Engineering

5.105

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-104 Probability, Statistics, and Random Processes for Electrical Engineering

5.106

© 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 5-105 Probability, Statistics, and Random Processes for Electrical Engineering

5.107

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-106 Probability, Statistics, and Random Processes for Electrical Engineering

5.108

© 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 5-107 Probability, Statistics, and Random Processes for Electrical Engineering

5.109

© 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 5-108 Probability, Statistics, and Random Processes for Electrical Engineering

5.9 Pairs of Jointly Gaussian Random Variables

5.110

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-109 Probability, Statistics, and Random Processes for Electrical Engineering

5.111

5.112

© 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 5-110 Probability, Statistics, and Random Processes for Electrical Engineering

5.113

© 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 5-111 Probability, Statistics, and Random Processes for Electrical Engineering

5.114

(a)

(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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-112 Probability, Statistics, and Random Processes for Electrical Engineering

(d)

(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 5-113 Probability, Statistics, and Random Processes for Electrical Engineering

(f)

(e)

© 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 5-114 Probability, Statistics, and Random Processes for Electrical Engineering

5.115

© 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 5-115 Probability, Statistics, and Random Processes for Electrical Engineering

5.116

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 5-116 Probability, Statistics, and Random Processes for Electrical Engineering

5.117

5.118

© 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 5-117 Probability, Statistics, and Random Processes for Electrical Engineering

5.119

© 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 5-118 Probability, Statistics, and Random Processes for Electrical Engineering

5.120

© 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 5-119 Probability, Statistics, and Random Processes for Electrical Engineering

5.10 Generating Independent Gaussian Random Variables The following Octave code produces the inverse: function z = rayleigh_rnd(s) u = rand; z = s.*(2.*log(1./(1-u))).^(1/2); end The following Octave generates the requested pairs and plot: len = 10000; X = discrete_rnd([1 -1], [0.5 0.5], 2, len); N = normal_rnd(0, 1, 2, len); Y = X + N; figure; plot(Y(1,:), Y(2,:), "."); Xr = sign(Y); sigerr = (X ~= Xr); biterr = (sigerr(1,:) | sigerr(2,:)); proberr = sum(biterr)./len The following Octave generates the requested pairs and plot: X = normal_rnd(0, 2, 1, 1000); N = normal_rnd(0, 1, 1, 1000); Y = X + N; Xr = Y./(1 + 1/2); err = Xr - X; figure; hist(err, [-3:0.25:3], 4); m = mean(err) v = var(err)

5.121

5.123

5.124

© 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 5-120 Probability, Statistics, and Random Processes for Electrical Engineering

The following Octave generates the sequence of Xn and Yn: X = normal_rnd(0, 1, 1, 1000); Y = (X + [0 X(1:999)])./2; figure; plot(Y(1:999), Y(2:1000), "."); Z = (X - [0 X(1:999)])./2; figure; plot(Z(1:999), Z(2:1000), "."); The following Octave generates the specified jointly Gaussian random variables: function vw = gaussian_correlate(xy, mu1, mu2, var1, var2, rho) sig1 = sqrt(var1); sig2 = sqrt(var2); K = [var1, rho*sig1*sig2; rho*sig1*sig2, var2]; [evec eval] = eig(K); A = evec * sqrt(eval); vw = zeros(size(xy)); for i = 1:length(xy) vw(:,i) = A*xy(:,i) + [mu1; mu2]; end end

5.125

5.126

© 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 5-121 Probability, Statistics, and Random Processes for Electrical Engineering

The following Octave generates the specified jointly Gaussian random variables and plot: xy = normal_rnd(0, 1, 2, 1000); vw = gaussian_correlate(xy, 1, -1, 1, 2, -1/2); plot(vw(1,:), vw(2,:), "."); The following Octave generates the specified plot: xy = normal_rnd(0, 1, 2, 1000); muh = 174; muv = 4.4; varh = 42.36; varv = 0.021; covhv = 0.458; sigh = sqrt(varh); sigv = sqrt(varv); rhohv = covhv/(sigh*sigv); xy = normal_rnd(0, 1, 2, 1000); hv = zeros(size(xy)); hv = gaussian_correlate(xy, muh, muv, varh, varv, rhohv); hw = zeros(size(hv)); hw(1,:) = hv(1,:); hw(2,:) = e.^hv(2,:); bmi = hw(2,:)./(hw(1,:).^2); hist(bmi);

5.127

5.128

© 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 5-122 Probability, Statistics, and Random Processes for Electrical Engineering

Problems Requiring Cumulative Knowledge

5.129

© 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 5-123 Probability, Statistics, and Random Processes for Electrical Engineering

© 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 5-124 Probability, Statistics, and Random Processes for Electrical Engineering

5.130

© 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 5-125 Probability, Statistics, and Random Processes for Electrical Engineering

5.131

© 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 5-126 Probability, Statistics, and Random Processes for Electrical Engineering

5.132

© 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 5-127 Probability, Statistics, and Random Processes for Electrical Engineering

5.133

© 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.