Continuous Uniform Random Variable¶
Continuous Uniform Random Variable¶
A uniform random variable has the following distribution function
\[
\begin{aligned}
f_{X}(x) = \begin{cases} \frac{1}{b-a} &\mbox{$if a \leq x \leq b$}\newline
0 &\mbox{otherwise} \end{cases}
\end{aligned}
\]
Mean and Variance¶
\[
\begin{aligned}
E[X] &= \int_{a}^{b} x \frac{1}{b-a} dx = [\frac{x^{2}}{2(b-a)}]\_{a}^{b}\newline
&= \frac{a+b}{2}\newline
Var(X) &= \int_{a}^{b} (x - \frac{a+b}{2})^{2} \frac{1}{b-a} dx \newline
&= \frac{(b-a)^{2}}{12}
\end{aligned}
\]
Moment Generating Function¶
\[
\begin{aligned}
E[e^{tX}] &= \int_{a}^{b} e^{tx} \frac{1}{b-a} dx\newline
&= \frac{e^{tb} - e^{ta}}{t(b-a)}
\end{aligned}
\]