Appendix

Probability Distributions

The uniform distribution on has the density

$f_{[a,b]}(x) = \{ \begin{array}{cc} \frac1{b-a} & a <= x <= b \\ 0 & {otherwise} \\ \end{array} .$

The normal distribution with mean and variance has the density

$f_{µ,sigma^2(x) = \frac1{\sqrt{2pi} * sigma} * e^{-\frac{(x-µ)^2{2sigma^2$

The gamma distribution with parameters has the density

$f_{a,lambda}(x) = \frac{lambda^{a}{Gamma(a)} * e^{-lambda * x} * x^{a-1} (x>0)$

A gamma distribution with parameters $(\fracn2,\frac12)$ is called -distribution with degrees of freedom, and a gamma distribution with parameters is called exponential distribution.

If and are independent random variables with and , then the distribution of $T = \fracU{\sqrt{V/n}$ is called t-distribution with degrees of freedom and has the density

$f_n(x) = \frac1{\sqrt{pi * n} * \frac{Gamma\......ht)} * ( 1 + \frac{x^2n )^{-\frac{n+1}2$

If and are independent random variables with $U ~ chi_{a}^2$ and $V ~ chi_{b}^2$ , then the distribution of $T = \frac{U/a}{V/b}$ is called F-distribution with degrees of freedom and has the density

$f_{a,b}(x) = \frac{a}{b} * \frac{Gamma(\frac{a+b}......rac{a}{b}x )^{\frac{a+b}2 {for} \; x>0$