What's a gamma distribution and how does one apply an mgf to it?

**Intsecxtanx** · October 23rd 2009, 05:20 PM

Apply the m.g.f. of the gamma distribution to find its expectation and variance. Any help will be very much appreciated thank you so much.

**matheagle** · October 23rd 2009, 05:29 PM

Do you mean OBTAIN the mgf of a gamma?
That's been done here before.
Just combine the exponential terms and observe the new beta (so it's a valid density)
and you're done.

**akolman** · October 23rd 2009, 06:01 PM

Well, if you are given the mgf, M(t), then
$E[X]=\frac{d}{dt}M(0)$
and
$Var(X)=E[X^2]-(E[X])^2=\frac{d^2}{dt^2}M(0)-(E[X])^2$

If you are not given the mgf, then you have to calculate
$M(t)=\int_{- \infty}^{\infty}e^{tx}f(x)dx$ where f(x) is the pdf of the gamma distribution

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