Page 285 - Fundamentals of Probability and Statistics for Engineers

268 Fundamentals of Probability and Statistics for Engineers

Let us define a new random variable Y by
n
X q ln f X j ; †
Y ˆ : 9:31†
q
jˆ1
Equation (9.30) shows that
EfYgˆ 0:

Moreover, since Y is a sum of n independent random variables, each with mean
2
zero and variance Ef[q ln f X; )/q ] g, the variance of Y is the sum of the n
variances and has the form
)
qlnf X; †
2
2
ˆ nE : 9:32†
Y q
Now, it follows from Equation (9.29) that

^
1 ˆ Ef Yg: 9:33†
Recall that

^
^
^
Ef Ygˆ Ef gEfYg‡ ^ Y Y ;
or
^
1 ˆ 0†‡ ^ Y Y : 9:34†
2
As a consequence of property 1, we finally have

1 1;
2
2
^ Y
or, using Equation (9.32),

2 )) 1
1 qlnf X; †
2
ˆ nE : 9:35†
^
2 q
Y
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

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