Finding probability function of moment generating function

Hi,

I have the following moment generating function:

$\displaystyle M_X(t) = 0.2e^{-3t}+0.11+0.1e^t+0.34e^{2t}+0.05e^{3t}+0.2e^{10t}$

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

$\displaystyle M_X(t) = \sum_{x \in X}e^{tx}f(x)$

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?

Re: Finding probability function of moment generating function

This seems to take advantage of fact that $\displaystyle e^{xt}$ are linearly independent. In order to get the term $\displaystyle .2 e^{-3t}$, what must happen? In particular, does this imply that $\displaystyle P(X = x) = .2$ for some $\displaystyle x$? Similar logic applies to each term in the summation.

Re: Finding probability function of moment generating function

I'm sorry, but I'm not quite sure I understand. I have a feeling that $\displaystyle x=-3$ would produce $\displaystyle 0.2$, but I can't justify it.

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.

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?

Re: Finding probability function of moment generating function

Effectively, yes. It means that, if $\displaystyle X$ has the distribution associated with that pmf then $\displaystyle X$ takes on the values you mentioned with probability 1.

Re: Finding probability function of moment generating function

Perfect. Thank you so much!