Page 365 - DMTH404_STATISTICS

2
Unit 25: Chi - Sqaure ( ) Distribution

25.2 Summary Notes

 We know that if X is a random variate distributed normally with mean m and standard
-
X m 2 (X m- ) 2
deviation s, then z = is a standard normal variate. Square of z, i.e., z =
s s 2
2
2 
if distributed as  - variate with one degree of freedom and is written as 1 . Further, the
2
value of  1 , a squared value, will lie between 0 to , for z lying between -  to . Since
2
most of the z-values are close to zero, the probability density of  will be highest near
2
zero. The  distribution with one degree of freedom is shown in Figure 25.1.
Generalising the above result, we can say that if X , X ...... X are n independent normal
1 2 n
variates each with mean m and standard deviations s , i = 1, 2, ...... n, respectively, then the
i i
(X - m ) 2 2
2 i i 2 
sum of squares å z = å is a  variate with n degrees of freedom, i.e., n .
i 2
s i
2
Thus, we can say that  n is sum of squares of n independent standard normal variate.
2
nS 2
 Since the random variable 2 is a  -variate with (n - 1) d.f.,
s

2
é nS ù n 2
-
( ) =
-
therefore E ê 2 ú = n 1 or 2 E S n 1 .
ë s û s
n 1
-
2 2
E S
Thus, we have ( ) = s ×
n
1 2 n
2 2 2
Further, if we define s = å ( X - X ) so that s = S × , we have
i
-
-
n 1 n 1
n 2 n n 1 2 2
-
2
( ) =
( ) =
E s × E S × s × = s (See Remarks 2 below).
-
n 1 n 1 n
-
25.3 Keywords
Standard normal variate: if X is a random variate distributed normally with mean m and
X m
-
standard deviation s, then z = is a standard normal variate.
s
Distribution: The distribution has only one parameter, i.e., number of degrees of freedom or d.f.
(in abbreviated form) which is a positive integer.

LOVELY PROFESSIONAL UNIVERSITY 357