Fundamentals of Probability and Statistics for Engineers

In principle, the distribution of S^2 can be derived with use of techniques
advanced in Chapter 5. It is, however, a tedious process because of the complex
nature of the expression for S^2 as defined by Equation (9.7). For the case in
which population X is distributed according to N(m,^2 ), we have the following
result (Theorem 9.1).

Theorem 9. 1: Let S^2 be the sample variance of size n from normal population
N(m,^2 ), then (n 1)S^2 /^2 has a chi-squared (^2 ) distribution with (n 1)
degrees of freedom.

Proof of Theorem 9.1:the chi-squared distribution is given in Section 7.4.2.
In order to sketch a proof for this theorem, let us note from Section 7.4.2 that
random variable Y,

has a ch i-squared distribution of n degrees of fr eedom since each term in the
sum is a squared normal random variable and is independent of other random
variables in the su m. Now, we can show that the difference between Y and
is

Since the right-hand side of Equation (9.13) is a random variable having a chi-
squared distribution with one degree of freedom, Equation (9.13) leads to the
result that (n 1)S^2 /^2 ischi-squared distributed with (n 1) degrees of freedom
provided that independence exists between (n 1)S^2 /^2 and

The proof of this independence is not given here but can be found in more
advanced texts (e.g. Anderson and Bancroft, 1952).

9.1.3 Sample M oments

The kth sample moment is

Parameter Estimation 263



    

Yˆ

1

^2

Xn

iˆ 1

…Xim†^2 ; … 9 : 12 †

n1)S^2 /^2

Y

…n 1 †S^2

^2

ˆ…Xm†



n^1 =^2

 1 2

: … 9 : 13 †

  



Xm)



n1/2

 1 2

Mkˆ

1

n

Xn

iˆ 1

Xik: … 9 : 14 †