Assume X a random variable with a finite variance, and also that $M_{X}\left(t\right)=at+b$, what is the distribution for X.
I really have no idea where to start, I know that $at+b=\sum_{x\in\mathbb{R}}e^{tx}P\left(X=x\right)$ so I thought that if $$P\left(X=0\right)=b\ \land\ P\left(X=\frac{ln\left(t\right)}{t}\right)=a$$ and for any other x P(X=x)=0 i get what i want, but that is only true if a+b=1 no? and I couldn't manage to prove that, and also I don't know how to use the fact that X has finite variance. any help is welcomed :)