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

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- Sep 26th 2008, 02:48 PMweakmathRandom Variable
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 - Sep 26th 2008, 03:34 PMmr fantastic
This thread contains the ideas you need to use: http://www.mathhelpforum.com/math-he...tml#post119542

- Sep 26th 2008, 03:39 PMweakmath
does it make a difference that your paremeters are unknown while solving and also, the result of subscript 1 and n?

- Sep 26th 2008, 03:49 PMmr fantastic
- Sep 26th 2008, 04:25 PMmr fantastic
- Sep 27th 2008, 05:28 AMmr fantastic
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, ....