A uniform random variable has the following distribution function \begin{align} f_{X}(x) = \begin{cases} \frac{1}{b-a} &\mbox{$if a \leq x \leq b$}\newline 0 &\mbox{otherwise} \end{cases} \end{align}
\begin{align} E[X] &= \int_{a}^{b} x \frac{1}{b-a} dx = [\frac{x^{2}}{2(b-a)}]_{a}^{b}\newline &= \frac{a+b}{2}\newline Var(X) &= \int_{a}^{b} (x - \frac{a+b}{2})^{2} \frac{1}{b-a} dx \newline &= \frac{(b-a)^{2}}{12} \end{align}
\begin{align} E[e^{tX}] &= \int_{a}^{b} e^{tx} \frac{1}{b-a} dx\newline &= \frac{e^{tb} - e^{ta}}{t(b-a)} \end{align}