1. Suppose that X ~ N (mu, sig^2). Show
((x-mu)/sig)^2 ~ Gamma(1/2, 1/2.) = Ki-square subscripte 1
2. Suppose that X1,X2.... Xm are iid N(mu, sig^2). Show that
summation i = 1 to n ( (Xi - mu)/sig)^2 ~ Ki-square subscript n
This thread contains the ideas you need to use: http://www.mathhelpforum.com/math-he...tml#post119542
In fact, at second glance Q1 is even easier than I thought. The thread I referenced actually gives a solution to Q1 since
~ .
Q2:
The moment generating function of is:
(which by the way is recognised as the moment generating function for the chi-squared distribution with 1 degree of freedom).
Therefore the moment generating function of
(since are i.i.d. random variables).
This is recognised as the moment generating function for the chi-squared distribution with n degrees of freedom. And since when a moment generating function exists there is a unique distribution corresponding to that moment generating function, ....