Chapter 6: Vector Random Variables - 기록용 블로그 - 기록용...

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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6.2 Functions of Several Random Variables

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6.3 Expected Values of Vector Random Variables

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6.4 Jointly Gaussian Random Vectors

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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.

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6.5 Estimation of Random Variables

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6.6 Generating Correlated Vector Random Variables

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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);

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Problems Requiring Cumulative Knowledge

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