Moment generating function of sum of exponential samples

[millwallcrazy]

I know that hte MGF is = the E[e^tx]

How do i show that if i take a sample (X1;X2; : : :Xn) from the exponential density f(x) = A*e^(-Ax), then the sum Z = sum(Xi) has the gamma density?

I found that the MGF for the exponential was A/(t-A) if that helps

Thanks

 

[statdad]

Note that

$$e^{t\sum X_i}} = \prod{e^{tX_i}}$$

This will relate the form of the mgf of the sum to the individual mgfs of the sample.