Finding probability function of moment generating function

• Jul 4th 2011, 12:21 PM
sorrofix
Finding probability function of moment generating function
Hi,

I have the following moment generating function:
$M_X(t) = 0.2e^{-3t}+0.11+0.1e^t+0.34e^{2t}+0.05e^{3t}+0.2e^{10t}$

I would like to find the support and the value of f_X(0) (i.e. the probability function of X evaluated at 0), but I am unsure how to go about doing this. I'm just guessing here, but couldn't X take on any value?

I also know that a moment generating function is defined as
$M_X(t) = \sum_{x \in X}e^{tx}f(x)$
So I factored e^t out of the original equation, but I'm not sure how to get the original equation into a summation in terms of x. Can someone help me understand this better?
• Jul 4th 2011, 12:55 PM
Guy
Re: Finding probability function of moment generating function
This seems to take advantage of fact that $e^{xt}$ are linearly independent. In order to get the term $.2 e^{-3t}$, what must happen? In particular, does this imply that $P(X = x) = .2$ for some $x$? Similar logic applies to each term in the summation.
• Jul 4th 2011, 01:00 PM
sorrofix
Re: Finding probability function of moment generating function
I'm sorry, but I'm not quite sure I understand. I have a feeling that $x=-3$ would produce $0.2$, but I can't justify it.
• Jul 4th 2011, 01:02 PM
Guy
Re: Finding probability function of moment generating function
Good. Now, do that for each term and use that to conjure up a pmf (probability mass function). Then, check that the pmf does, indeed, have that mgf.
• Jul 4th 2011, 03:56 PM
sorrofix
Re: Finding probability function of moment generating function
Ahh, that makes sense. So does that mean X can only take on the values -3,0,1,2,3,10?
• Jul 4th 2011, 03:59 PM
Guy
Re: Finding probability function of moment generating function
Effectively, yes. It means that, if $X$ has the distribution associated with that pmf then $X$ takes on the values you mentioned with probability 1.
• Jul 4th 2011, 04:03 PM
sorrofix
Re: Finding probability function of moment generating function
Perfect. Thank you so much!