Suppose that the random variable Y has the moment generating function:
$M_y(t) = \frac{1}{5}(e^{-2t} + e^{-t} + 1 + e^t + e^{2t})$
for all real t.
1) Find $E[Y]$ and $VAR[Y]$.
2) Give a general formula for calculating $E[Y^n]$ for all n.
For problem 1, I took the derivative of the mgf and set t=$0$ and got $E[Y] = 0$ and then I took the second derivative and got $VAR[Y] = 2$.
Then for problem 2, I tried taking the derivative a couple more times and what I got was:
$M_{y^n}(t) = \frac{1}{5}((-1)^n2n + (-1)^n + 1 + 2n)$. This seemed to work whenever $t=0$ but I may be wrong. Any insight is greatly appreciated.