Given $E(X^r)$ find MGF and PMF

Question

If $E(X^r) = 5^r, r = 1,2,3, \ldots $, how would you find the moment generating function of $X$ and the PMF of $X$?

So far I have$$ M(t) = M(0) + \sum_{i=0}^\infty 5^r \frac{t^r}{r!}.$$ Am I doing it right?

Answer 1

$$M(t) = M(0) + \sum_{\color{red}{r=1}}^\infty E[X^r]\frac{t^r}{r!}=\sum_{r=0}^\infty \frac{(5t)^r}{r}=\exp(5t)$$

• From the MGF, try to identify the PMF.