Reversing the moment generating function

#1
So this is my moment generating function

\( \frac{e^{-2t}+e^{-t}+e^{t}+e^{2t}}{4} \)

Now i need to find the PDF or CDF

So i think the moment genereting function needs to be divided by \(e^{tx} \)

But how do i do this in an algabraic way
This is what i got so far

\(\frac{e^{-2t}*e^{tx}+e^{-t}*e^{tx}+e^{t}*e^{tx}+e^{2t}*e^{tx}}{4}\)

And now i'm stuck :(
 

BGM

TS Contributor
#2
Note that the mgf of a random variable \( X \) is defined by

\( M_X(t) = E[e^{tX}] \)

and mathematically speaking it is a Laplace transform of the corresponding pmf/pdf. Now you are given the mgf, and want to find the corresponding pmf/pdf, so generally speaking you would like to do an inverse-Laplace transform. There are table for this for many common functional form.

However, your mgf is easy enough to recognize. Think about how do you calculate the mgf for a discrete distribution, say a Binomial distribution. Then you can match the corresponding probabilities and support points. Also note that the Laplace transform is unique so you will have a unique solution.