Question

Prove the following theorem: Let X ~ χ, and Y ~ χ21. If X and Y are independent, then

Answer 1

For,

$X \sim \chi^2_n$

We have the Moment generating function as :

$M_X(t)=(1-2t)^{-n/2}$

Similarly,

$Y \sim \chi^2_m$

Moment generating function will be :

$M_Y(t)=(1-2t)^{-m/2}$

Now,

The moment generating function of a sum of mutually independent random variables is just the product of their moment generating functions

Moment generating function of X+Y is defined as :

$M_{X+Y}(t) = M_X(t).M_Y(t)$

$=(1-2t)^{-n/2}.(1-2t)^{-m/2}$

$=(1-2t)^{-(n+m)/2}$

The above moment generating function is of Chi square distribution with degrees of freedom (n+m)

Hence, proved.