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!
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?
Last edited by kastchei2112; 11-21-2010 at 05:31 PM. Reason: Trimming quote.
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.
Tweet |