
Problem:
I have a random variable $\displaystyle U,$ which has a uniform distribution with parameters 0 and 2. I have another random variable $\displaystyle X$, which has a moment generating function of $\displaystyle {(1ux)^{1}}$ whereby $\displaystyle U=u$.
1. How do I find E(X)? Do I simply differentiate the mgf? [SOLVED]
2. How do I find $\displaystyle E(XU)$ and $\displaystyle Var(XU)$?
Your help is very much appreciated.

I know that the mgf is an exponential distribution with parameters 1, and mean $\displaystyle u$ and variance $\displaystyle u^2$ (silly me). I solved them by differentiating the mgf, and then substituting in 0, and then obtaining the variance through the necessary equation. But I still confused about how to get $\displaystyle E(XU)$ and $\displaystyle Var(XU)$. Can anyone help me please?