# Thread: moment generating function problem

1. ## moment generating function problem

Hi everyone, I'm stuck on a probability problem and was wondering if anyone could help me get started? I would really appreciate it!

Let X,Y,Z be random variables, either continuous or discrete. The joint MGF of X,Y,Z is defined by

Mxyz(t1,t2,t3) = Ee^(t1(X)+t2(Y)+t3(Z))

Show that:

Thanks in advance to anyone who can give me a hint as to what I should be looking up to try and do this on my own!

2. ## Re: moment generating function problem

I guess just follow the definition is ok.

Hence

3. ## Re: moment generating function problem

Originally Posted by BGM

It looks like I am on the same problem.

Question 1: Are you able to swap the Expected Value with the differentiation (quoted) because the transforms for expected value don't include any of the t's? It's not readily intuitive that

Question 2: This regards a previous part of the same problem. We are to show that

I can easily do this if I assume that X, Y, and Z are also independent, but this wasn't given in the problem. I can see that

This seems like Mx+y+z, but how do I make sense of the actual integrals? It seems like

which doesn't intuitively seem to equal

Am I over-thinking this?

4. ## Re: moment generating function problem

http://en.wikipedia.org/wiki/Leibniz_integral_rule

Yes I am quite lazy the check the technical condition.

You need the Fubini Theorem/Dominated Convergence Theorem
to exchange the order of limit.

But I guess in most of the case, when you required to use the moment
generating function to generate the moment, you need your mgf to satisfy
such a technical condition; otherwise, differentiating the mgf may not
yield the desired result.

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