Page 126 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 109

If random variables X and Y are independent, then we also have
XY t; s† ˆ X t† Y s†: 4:86†

To show the above, we simply substitute f (x)f (y) for f XY (x, y) in Equation
X

Y
(4.83). The double integral on the right-hand side separates, and we have
Z 1 Z 1
jtx
jsy
XY t; s†ˆ e f x†dx e f y†dy
Y
X
1 1
ˆ X t† Y s†;
and we have the desired result.
Analogous to the one-random-variable case, joint characteristic function
XY (t, s) is often called on to determine joint density function f XY (x, y) of X
and Y and their joint moments. The density function f XY (x, y) is uniquely
determined in terms of XY (t, s) by the two-dimensional Fourier transform

1 Z 1 Z 1 j tx‡sy†
f x; y†ˆ e XY t; s†dtds; 4:87†
XY 2
4 1 1
n
m
and moments EfX Y gˆ nm , if they exist, are related to XY (t, s) by
q n‡m n‡m Z 1 Z 1 n m
XY t; s† ˆ j x y f XY x; y†dxdy
n
qt qs m
t;sˆ0 1 1 4:88†
ˆ j n‡m nm :

The MacLaurin series expansion of XY (t, s) thus takes the form

1 1
X X ik i k
XY t; s†ˆ jt† js† : 4:89†
i!k!
iˆ0 kˆ0
The above development can be generalized to the case of more than two
random variables in an obvious manner.
Example 4.18. Let us consider again the Brownian motion problem discussed
0
0
in Example 4.17, and form two random variables X and Y as
0
X ˆ X 1 ‡ X 2 ‡ ‡ X 2n ; )
4:90†
0
Y ˆ X n‡1 ‡ X n‡2 ‡ ‡ X 3n :

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