Thread: Gamma Function Proofs - having trouble

I feel like I should be understanding these, but just can't make any headway:

1) If X~gamma(alpha, theta), then show that cX~gamma(alpha, theta/c).

and

2) If X1, X2, ..., Xn are independent and ~Normal(mu, sigma^2), show that (1/sigma^2)(X1^2 + X2^2 + ... + Xn^2)~gamma(n/2, 1/2).

Any suggestions or help much appreciated.

Originally Posted by 1of42

I feel like I should be understanding these, but just can't make any headway:

1) If X~gamma(alpha, theta), then show that cX~gamma(alpha, theta/c). Mr F says: This is wrong. ~ .

[snip]

There are several aproaches. The easiest is to calculate the moment generating function of cX.

Note that if Y = cX and is the moment generating function of X then .

Of related interest: http://www.mathhelpforum.com/math-he...-function.html


Another approach is to calculate .

Then the pdf of Y is

Originally Posted by 1of42

[snip]
2) If X1, X2, ..., Xn are independent and ~Normal(mu, sigma^2), show that (1/sigma^2)(X1^2 + X2^2 + ... + Xn^2)~gamma(n/2, 1/2).

Any suggestions or help much appreciated.

Here is the outline of one possible approach:

Read this to get an idea of how to find the distribution of : http://www.mathhelpforum.com/math-he...tml#post119542

Use this to get (or look up) the moment generating function of .

Hence get the moment generating function of . Now identify this moment generating as being the same as that for the gamma distribution.


Threads of related interest:

http://www.mathhelpforum.com/math-he...a-samples.html

http://www.mathhelpforum.com/math-he...tribution.html

http://www.mathhelpforum.com/math-he...-function.html