Thread: Moment Generating Function of a gamma distribution.

Hello Math Forums, this is my first time posting here so I apologize for not knowing all the functions of the board, but I just have a stats question that I am a bit lost on.

I am supposed to find the moment generating function of a gamma distribution with parameters α,[SIZE=2]β. Then, obtain the first two moments and use them to find the expected value and variance.

I know that the mgf of a gamma distribution is:
M(t)= (1-β)^-α

But from there I'm not sure how to generate the first two moments or find the expected value or variance. I would greatly appreciate any help on this. Thank you.

Originally Posted by Yardee

Hello Math Forums, this is my first time posting here so I apologize for not knowing all the functions of the board, but I just have a stats question that I am a bit lost on.

I am supposed to find the moment generating function of a gamma distribution with parameters α,[SIZE=2]β. Then, obtain the first two moments and use them to find the expected value and variance.

I know that the mgf of a gamma distribution is:
M(t)= (1-β)^-α

But from there I'm not sure how to generate the first two moments or find the expected value or variance. I would greatly appreciate any help on this. Thank you.

First of all, the mgf of a gamma function is M(t)= (1-βt)^-α ; for β<(1/α)

You forgot to include "t" in the mgf.

To find the expected value, take the derivative of the mgf of the gamma function with respect to "t"(d M(t)/dt), and calculate its value at t=0. This will be E[X] or your expected value as well as the first moment

Then take the second derivative of the mgf of the gamma function with respect to t (d(^2)t/dt^2), and calculate this value at t=0. This will be E[(X)^2] your second moment.

Your variance can be calculated as V(X) = E[(X)^2] - (E[X])^2

or the variance = (the second moment) - (the first moment)^2

You should get : expected value = αβ

Variance = α(β^2)