Fundamentals of Probability and Statistics for Engineers

Let us define a new random variable Y by

Equation (9.30) shows that

Moreover, since Y is a sum of n independent random variables, each with mean
zero and variance the variance of Y is the sum of the n
variances and has the form

N ow, it follows from Equation (9.29) that

R ecall that

or

As a consequence of property^2 1, we finally have

or, using Equation (9.32),

The proof is now complete.

In the above, we have assumed that differentiation with respect to under an
integral or sum sign are permissible. Equation (9.26) gives a lower bound on the

268 Fundamentals of Probability and Statistics for Engineers

Yˆ

Xn

jˆ 1

qlnf…Xj;†

q

: … 9 : 31 †

EfYgˆ 0 :

Ef[qlnfX;)/q]^2 g,

^2 YˆnE

qlnf…X;†

q

) 2

: … 9 : 32 †

1 ˆEf^Yg: … 9 : 33 †

Ef^YgˆEf^gEfYg‡^Y^Y;

1 ˆ… 0 †‡^Y^Y: … 9 : 34 †

 

1

^2 ^^2 Y

 1 ;

^2 ^

1

^2 Y

ˆ nE

qlnf…X;†

q

)) 2  1

: … 9 : 35 †