Find moment generating function with specified pdf

Question

$f(x) = 2x/c^2$, $0<x<c$

Integration of $2xe^{tx}/c^2$ between 0 and c gives the MGF;

$2e^{tc}/c$.

But it does not seem right. I cant use it to find expected value. What is wrong?

Answer 1

This is a triangular distribution with pameters $a=0$ and $b=c$.

By direct integration $$ M_X(t)=\Bbb E(\mathrm e^{tX})=\frac{2}{c^2}\int_0^c x\,\mathrm e^{tx}\mathrm dx=\tfrac{2}{c^2}\frac{\mathrm d}{\mathrm dt}\left(\int_0^c \mathrm e^{tx}\mathrm dx\right)=\tfrac{2}{c^2}\frac{\mathrm d}{\mathrm dt}\left(\tfrac{\mathrm e^{ct}-1}{t}\right)=2\frac{\mathrm e^{ct}(ct-1)+1}{t^2c^2} $$ and the Taylor expansion using the expansion for $\mathrm e^{ct}$ is $$ M_X(t)=1+\frac{2c}{3}t+\frac{c^2}{2}\frac{t^2}{2}+o(t^3)=1+M_X'(0)t+M''_X(0)\frac{t^2}{2}+o(t^3) $$ and then $\Bbb E(X)=\frac{2c}{3}$