Page 280 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 263
2
In principle, the distribution of S can be derived with use of techniques
advanced in Chapter 5. It is, however, a tedious process because of the complex
2
nature of the expression for S 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).
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Theorem 9.1: Let S be the sample variance of size n from normal population

2
2
N(m, 2 ), then (n 1)S / 2 has a chi-squared ( ) 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 ,
n
1 X 2
Y ˆ X i m† ; 9:12†
2
iˆ1
has a chi-squared distribution of n degrees of freedom since each term in the
sum is a squared normal random variable and is independent of other random
variables in the sum. Now, we can show that the difference between Y and
2
n 1)S / 2 is
n 1†S 2 1 2
Y ˆ X m† : 9:13†
2 n 1=2
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
2
result that (n 1)S / is chi-squared distributed with (n 2 1) degrees of freedom
2
provided that independence exists between (n 1)S / 2 and
2

1
X m)
n 1/2
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 MOMENTS

The kth sample moment is

n
1 X k
M k ˆ X : 9:14†
i
n
iˆ1

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