## Tuesday, September 21, 2010

### Some Continuous Random Variables

From Introduction to Probability.

Continuous Uniform Over [a,b]:

$f_X(x)$ = $\frac{1}{b-a}$ , if $a \leq x \leq b$ or 0 otherwise

E[X] = $\frac{a+b}{2}$
var(X) = $\frac{(b-a)^2}{12}$

Exponential with Parameter $\lambda$:

$f_X(x)$ = $\lambda e^{-\lambda x}$, if $x \geq 0$ or 0 otherwise

$F_X(x)$ = $1 - e^{-\lambda x}$, if $x \geq 0$ or 0 otherwise

E[X] = $\frac{1}{\lambda}$
var(x) = $\frac{1}{{\lambda}^2}$

Normal with Parameters $\mu$ and ${\sigma}^2 > 0$:

$f_X(x)$ = $\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{{(x-\mu)}^2}{2{\sigma}^2}}$

E[X] = $\mu$
var(X) = ${\sigma}^2$