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Statistics

Notes 4. Additive property

2 2
The sum of two independent  variates is also a  variate with degrees of freedom
equal to the sum of their individual degrees of freedom.

 2  2
If n and m are two independent random variates with n and m degrees of freedom
2
 +  2  2
respectively, then n m is also a variate with n + m degrees of freedom.
Remarks:
1. The degrees of freedom is defined as the number of independent random variables. If n is
the number of variables and k is the number of restrictions on them, the degrees of
freedom are said to be n - k.
2. On the basis of the definition of degrees of freedom, given above, we can say that
2
n æ X - X ö 2
i
å ç ÷ is a  variate with (n - 1) degrees of freedom. It may be pointed out here
i 1è s ø
=
that one degree of freedom is reduced because for a given value of X , the number of
independent variables is (n - 1).
25.1.1 Sampling Distribution of Variance

 2 2 1 2
Using -distribution, we can construct the sampling distribution of S = å ( X - X ) .
i
n
Let X , X ...... X be a random sample of size n from a normal population with mean m and
1 2 n
2
variance s . We can write
X - m = ( X - X ) ( X m+ - )
i i
Squaring both sides and taking sum over all the n observations, we get

n 2 n 2 n 2 n
å (X - ) m = å ( X - X ) + å ( X - ) m + 2 å ( X - X )( X - ) m
i
i
i
i 1 i 1 i 1 i 1
=
=
=
=
n 2 2 n
å
= å ( X - X ) + ( n X m- ) + 2 ( X m- ) ( X - X )
i
i
=
=
i 1 i 1
We note that the last term is zero. Therefore, we have
n 2 n 2 2
å (X - ) m = å ( X - X ) + ( n X - ) m
i
i
=
=
i 1 i 1
2
Dividing both sides by s , we get
n 2 n 2
å (X - ) m å ( X - X ) n ( X m- ) 2
i
i
i 1 i 1
=
=
2 = 2 + 2
s s s
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