Thread: Exponentially Distributed RV - MGF

I need help. The problem is this:

X is an exponentially distributed random variable with p.d.f.

f(x) =
0<x<infinity

Show that the m.g.f. of X is for , and use this to find E(X) and Var(X).


I have no idea what to do with this problem or where to start. Please help point me somewhere? The only thing I could think of to potentially do is to undo the p.d.f. to get the original random variable, but...yeah. I'm really, really lost.

Originally Posted by notgod

I need help. The problem is this:

X is an exponentially distributed random variable with p.d.f.

f(x) =
0<x<infinity

Show that the m.g.f. of X is for , and use this to find E(X) and Var(X).


I have no idea what to do with this problem or where to start. Please help point me somewhere? The only thing I could think of to potentially do is to undo the p.d.f. to get the original random variable, but...yeah. I'm really, really lost.

just combine the exponential terms.



You can now integrate BUT it's better to notice another exponential density





since that is another valid density, just let

Let X and Y be two Statistically independent Exponentially distributive random variables with parameters and respectively.

a) find the CDF (commulative density function) of Z = X/X+Y , (use property of exponential pdf to evaluate the integrals in this question).

b) find the expected value of Z when =1, and = 2


So please can any one solve this, with all the steps of integration , Thanks a lot.

I think Laurent solved this a couple of days ago here.
Z is a beta.