Find moment generating function given rth moment without Laplace

Question

Let W be a random variable.

Given the rth moment, $E(W^{r}) = \frac{r!+6^{r}}{2^{r}} = M_w^{(r)}(0)$, $r \geq 0$,

how does one derive $f(w)$?

I know that

$M_w(t)=E(e^{wt}) = \begin{cases} \int_{-\infty}^{\infty} e^{wt} f(w) dw & \text{ if W is continuous} \\ \sum_{w = -\infty}^{\infty} e^{wt} f(w) dw & \text{ if W is discrete} \end{cases}$.

So, I can obviously compute the rth moments and equate them to the rth moments I get from differentiating the arbitrary mgf, but I don't see how that's helpful...

And how do we know if W is discrete or continuous?

And no using Laplace transforms. :P

Answer 1

Never mind I got it. One has to expand $e^{tx}$.

http://en.wikipedia.org/wiki/Moment-generating_function#Calculation