I have a random variable $C$ with probability density $\frac{c}{8}$ where $0\leq c \leq 4$. I am having trouble finding the moment generating function of this distribution.
$\begin{split} M_C(t) &= E_C(e^{tc})\\ &= \int_{0}^{4}e^{tc}\frac{c}{8}\,dc\\ &= \frac{1}{8}\int_0^4e^{tc}c\,dc\end{split}$
This is as far as I can get. I think that I have to somehow manipulate the expression in the integral to become a distribution, causing the integral to evaluate to one. I will then be left with some function of $t$ on the outside of the integral.
Can anyone help me find this MGF? Anything would be much appreciated.