Fundamentals of Probability and Statistics for Engineers

and, from Equations (4.23) and (4.24),

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

We can also show, following a similar procedure, that

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

where pX 1 ...Xn and fX 1 ...Xnare, respectively, the joint mass function and joint
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. LetXbe

92 Fundamentals of Probability and Statistics for Engineers

ˆ 0 :

jj1:

^211 ˆj 11 j^2  20  02 : … 4 : 30 †

E^2 fXYgˆjEfXYgj^2 EfX^2 gEfY^2 g: … 4 : 31 †

Efg…X 1 ;...;Xn†gˆ

X

i 1

...

X

in

g…x 1 i 1 ;...;xnin†pX 1 ...Xn…x 1 i 1 ;...;xnin†;

X 1 ;...;Xndiscrete; … 4 : 32 †

Efg…X 1 ;...;Xn†gˆ

Z 1

1

...

Z 1

1

g…x 1 ;...;xn†fX 1 ...Xn…x 1 ;...;xn†dx 1 ...dxn;

X 1 ;...;Xncontinuou s;… 4 : 33 †