Finding probability function of moment generating function

Hi,

I have the following moment generating function:

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

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 are linearly independent. In order to get the term , what must happen? In particular, does this imply that for some ? 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 would produce , 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 has the distribution associated with that pmf then takes on the values you mentioned with probability 1.

Re: Finding probability function of moment generating function

Perfect. Thank you so much!