Gamma Function Proofs - having trouble

• Nov 11th 2008, 11:36 PM
1of42
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.
• Nov 12th 2008, 12:14 AM
mr fantastic
Quote:

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. ${\color{red}cX}$ ~ ${\color{red}\text{gamma}(\alpha, c \theta)}$.

[snip]

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

Note that if Y = cX and $m_X(t)$ is the moment generating function of X then $m_Y(t) = m_X(ct)$.

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

Another approach is to calculate $F(u) = \Pr(Y < u) = \Pr(cX < u) = \Pr\left( X < \frac{u}{c}\right)$.

Then the pdf of Y is $f(u) = \frac{dF}{du}$
• Nov 12th 2008, 12:35 AM
mr fantastic
Quote:

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 $Y_i = \frac{1}{\sigma^2} \, X_i^2$: http://www.mathhelpforum.com/math-he...tml#post119542

Use this to get (or look up) the moment generating function of $Y_i = \frac{1}{\sigma^2} \, X_i^2$.

Hence get the moment generating function of $\sum_{i = 1}^{n} Y_i$. Now identify this moment generating as being the same as that for the gamma distribution.