A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-1 Probability, Statistics, and Random Processes for Electrical Engineering

Chapter 6: Vector Random Variables

6.1 Vector Random Variables

6.1

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-2 Probability, Statistics, and Random Processes for Electrical Engineering

6.2

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-3 Probability, Statistics, and Random Processes for Electrical Engineering

6.3

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-4 Probability, Statistics, and Random Processes for Electrical Engineering

6.4

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-5 Probability, Statistics, and Random Processes for Electrical Engineering

(b)

6.5

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-6 Probability, Statistics, and Random Processes for Electrical Engineering

(c)

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-7 Probability, Statistics, and Random Processes for Electrical Engineering

6.6

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-8 Probability, Statistics, and Random Processes for Electrical Engineering

6.7

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-9 Probability, Statistics, and Random Processes for Electrical Engineering

6.8

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-10 Probability, Statistics, and Random Processes for Electrical Engineering

6.9

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-11 Probability, Statistics, and Random Processes for Electrical Engineering

6.10

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-12 Probability, Statistics, and Random Processes for Electrical Engineering

6.11

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-13 Probability, Statistics, and Random Processes for Electrical Engineering

6.12

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-14 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-15 Probability, Statistics, and Random Processes for Electrical Engineering

6.13

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-16 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-17 Probability, Statistics, and Random Processes for Electrical Engineering

6.14

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-18 Probability, Statistics, and Random Processes for Electrical Engineering

6.15

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-19 Probability, Statistics, and Random Processes for Electrical Engineering

6.16

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-20 Probability, Statistics, and Random Processes for Electrical Engineering

6.17

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-21 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-22 Probability, Statistics, and Random Processes for Electrical Engineering

6.2 Functions of Several Random Variables

6.18

6.19

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-23 Probability, Statistics, and Random Processes for Electrical Engineering

6.20

6.21

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-24 Probability, Statistics, and Random Processes for Electrical Engineering

6.22

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-25 Probability, Statistics, and Random Processes for Electrical Engineering

6.23

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-26 Probability, Statistics, and Random Processes for Electrical Engineering

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-27 Probability, Statistics, and Random Processes for Electrical Engineering

6.24

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-28 Probability, Statistics, and Random Processes for Electrical Engineering

6.25

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-29 Probability, Statistics, and Random Processes for Electrical Engineering

6.26

6.27

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-30 Probability, Statistics, and Random Processes for Electrical Engineering

6.28

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-31 Probability, Statistics, and Random Processes for Electrical Engineering

6.29

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-32 Probability, Statistics, and Random Processes for Electrical Engineering

6.3 Expected Values of Vector Random Variables

6.30

6.31

6.32

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-33 Probability, Statistics, and Random Processes for Electrical Engineering

6.33

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-34 Probability, Statistics, and Random Processes for Electrical Engineering

6.34

(a)

(b)

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-35 Probability, Statistics, and Random Processes for Electrical Engineering

6.35

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-36 Probability, Statistics, and Random Processes for Electrical Engineering

6.36

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-37 Probability, Statistics, and Random Processes for Electrical Engineering

6.37

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-38 Probability, Statistics, and Random Processes for Electrical Engineering

6.38

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-39 Probability, Statistics, and Random Processes for Electrical Engineering

6.39

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-40 Probability, Statistics, and Random Processes for Electrical Engineering

6.40

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-41 Probability, Statistics, and Random Processes for Electrical Engineering

6.41

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-42 Probability, Statistics, and Random Processes for Electrical Engineering

6.42

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-43 Probability, Statistics, and Random Processes for Electrical Engineering

6.43

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-44 Probability, Statistics, and Random Processes for Electrical Engineering

b)

6.44

6.45

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-45 Probability, Statistics, and Random Processes for Electrical Engineering

6.47

6.46

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-46 Probability, Statistics, and Random Processes for Electrical Engineering

6.48

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-47 Probability, Statistics, and Random Processes for Electrical Engineering

6.49

6.50

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-48 Probability, Statistics, and Random Processes for Electrical Engineering

6.51

6.52

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-49 Probability, Statistics, and Random Processes for Electrical Engineering

6.53

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-50 Probability, Statistics, and Random Processes for Electrical Engineering

6.4 Jointly Gaussian Random Vectors

6.54

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-51 Probability, Statistics, and Random Processes for Electrical Engineering

6.55

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-52 Probability, Statistics, and Random Processes for Electrical Engineering

6.56

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-53 Probability, Statistics, and Random Processes for Electrical Engineering

6.57

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-54 Probability, Statistics, and Random Processes for Electrical Engineering

6.58

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-55 Probability, Statistics, and Random Processes for Electrical Engineering

6.59

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-56 Probability, Statistics, and Random Processes for Electrical Engineering

If we write out the quadratic form in the exponent we obtain: 1 1

1 1 1 1

12

1

12

1

( )( ) ( )( )

= ( ) 2( ) ( )

1 ( ) 2( ) ( )

n n n n

jk j j k k jk j j k kj k j k

n

nn n n n n jk j jj

n

nn n n n n jk j jjnn

Q x m x m Q x m x m

Q x m x m Q x m

Q x m x m Q x mQ

− −

= = = =

−

=

−

=

− − − − −

− + − −

⎧ ⎫= − + − −⎨ ⎬

⎩ ⎭

∑ ∑ ∑ ∑

∑

∑

{ }{ }

2 2 2

2 2

1

1

( ) 2( )

= ( )

1 where ( )

nn n n n n nn

nn n n nn

n

jk j jjnn

Q x m x m B B Q B

Q x m B Q B

B Q x mQ

−

=

= − + − + −

− + −

= −∑

In the first line, all the terms in the second summation are contained in the first summation. We then completed the square to obtain an expression involving xk, its mean mk – B, and its variance 1/Qkk. The term QkkB2 is part of the normalization constant. We therefore conclude that:

21

1

1 1

1 1exp ( )2

( | ,..., )2n

n

n jk j jjnn nn

X n nnn

x m Q x mQ Q

f x x xQπ

−

=

−

⎧ ⎫⎛ ⎞⎪ ⎪− − + −⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭=

∑

We see that the conditional mean of xn is a linear function of the observations x1, x2,…, xn-1.

6.60

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-58 Probability, Statistics, and Random Processes for Electrical Engineering

6.63

6.64

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-61 Probability, Statistics, and Random Processes for Electrical Engineering

6.5 Estimation of Random Variables

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(a)

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-70 Probability, Statistics, and Random Processes for Electrical Engineering

(b)

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-73 Probability, Statistics, and Random Processes for Electrical Engineering

(a)

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-81 Probability, Statistics, and Random Processes for Electrical Engineering

6.77

6.78

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6.79

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-83 Probability, Statistics, and Random Processes for Electrical Engineering

6.6 Generating Correlated Vector Random Variables

6.80

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6.81

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6.82

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6.83

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-87 Probability, Statistics, and Random Processes for Electrical Engineering

The following Octave code generates a sequence of 1000 samples: X = normal_rnd(0, 1, 1, 1000); Y = (X + [0 X(1:length(X)-1)])./2; cov(Y);

6.84

(b)

(a)

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6.85

6.86

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6.87

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6.88

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Problems Requiring Cumulative Knowledge

6.89

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL 6-96 Probability, Statistics, and Random Processes for Electrical Engineering

b)

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6.91

6.92

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6.93

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