Page 109 - Fundamentals of Probability and Statistics for Engineers

92 Fundamentals of Probability and Statistics for Engineers

and, from Equations (4.23) and (4.24),
ˆ 0:

This is a simple example showing that X and Y are uncorrelated but they are
completely dependent on each other in a nonlinear way.

4.3.2 SCHWARZ INEQUALITY

In Section 4.3.1, an inequality given by Equation (4.25) was established in the
process of proving that j j 1:
2
2 ˆj 11 j 20 02 : 4:30†
11
We can also show, following a similar procedure, that

2
2
2
2
E fXYgˆjEfXYgj EfX gEfY g: 4:31†
Equations (4.30) and (4.31) are referred to as the Schwarz inequality. We point
them out here because they are useful in a number of situations involving
moments in subsequent chapters.

4.3.3 THE CASE OF THREE OR MORE RANDOM VARIABLES

The expectation of a function g(X 1 , X 2 ,..., X n ) of n random variables
X 1 , X 2 ,..., X n is defined in an analogous manner. Following Equations (4.18)
and (4.19) for the two-random-variable case, we have

X X
Efg X 1 ; ... ; X n †g ˆ ... †p †;
g x 1i 1
; ... ; x ni n
X 1 ...X n x 1i 1 ; ... ; x ni n
i 1 i n
X 1 ; ... ; X n discrete; 4:32†
Z 1 Z 1
Efg X 1 ; .. . ; X n †g ˆ ... g x 1 ; .. . ; x n †f x 1 ; ... ; x n † dx 1 ... dx n ;
X 1 ...X n
1 1
X 1 ; ... ; X n continuous; 4:33†
where p and f are, respectively, the joint mass function and joint
X 1 ...X n X 1 ...X n
density function of the associated random variables.
The important moments associated with n random variables are still the
individual means, individual variances, and pairwise covariances. Let X be

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