# 14.5 Formulas

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This module provides an overview of Statistics Formulas used as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.
Formula

## Factorial

$n!=n\left(n-1\right)\left(n-2\right)...\left(1\right)\text{}$

$0!=1\text{}$

Formula

## Combinations

$\left(\genfrac{}{}{0}{}{n}{r}\right)=\frac{n!}{\left(n-r\right)!r!}\text{}$

Formula

## Binomial distribution

$X~B\left(n,p\right)$

$P\left(X=x\right)=\left(\genfrac{}{}{0}{}{n}{x}\right){p}^{x}{q}^{n-x}$ , for $x=0,1,2,...,n$

Formula

## Geometric distribution

$X~G\left(p\right)$

$P\left(X=x\right)={q}^{x-1}p$ , for $x=1,2,3,...$

Formula

## Hypergeometric distribution

$X~H\left(r,b,n\right)$

$P\left(X=x\right)=\left(\frac{\left(\genfrac{}{}{0}{}{r}{x}\right)\left(\genfrac{}{}{0}{}{b}{n-x}\right)}{\left(\genfrac{}{}{0}{}{r+b}{n}\right)}\right)$

Formula

## Poisson distribution

$X~P\left(\mu \right)$

$P\left(X=x\right)=\frac{{\mu }^{x}{e}^{-\mu }}{x!}$

Formula

## Uniform distribution

$X~U\left(a,b\right)$

$f\left(X\right)=\frac{1}{b-a}$ , $a

Formula

## Exponential distribution

$X~\mathrm{Exp}\left(m\right)$

$f\left(x\right)=m{e}^{-\mathrm{mx}}$ , $m>0,x\ge 0$

Formula

## Normal distribution

$X~N\left(\mu ,{\sigma }^{2}\right)$

$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{{-\left(x-\mu \right)}^{2}}{{2\sigma }^{2}}}$ , $\phantom{\rule{12pt}{0ex}}-\infty

Formula

## Gamma function

$\Gamma \left(z\right)={\int }_{0}^{\infty }{x}^{z-1}{e}^{-x}\mathrm{dx}$ $z>0$

$\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$

$\Gamma \left(m+1\right)=m!$ for $m$ , a nonnegative integer

otherwise: $\Gamma \left(a+1\right)=\mathrm{a\Gamma }\left(a\right)$

Formula

## Student-t distribution

$X~{t}_{\mathrm{df}}$

$f\left(x\right)=\frac{{\left(1+\frac{{x}^{2}}{n}\right)}^{\frac{-\left(n+1\right)}{2}}\Gamma \left(\frac{n+1}{2}\right)}{\sqrt{\mathrm{n\pi }}\Gamma \left(\frac{n}{2}\right)}$

$X=\frac{Z}{\sqrt{\frac{Y}{n}}}$

$Z~N\left(0,1\right)$ , $Y~{Χ}_{\mathrm{df}}^{2}$ , $n$ = degrees of freedom

Formula

## Chi-square distribution

$X~{Χ}_{\mathrm{df}}^{2}$

$f\left(x\right)=\frac{{x}^{\frac{n-2}{2}}{e}^{\frac{-x}{2}}}{{2}^{\frac{n}{2}}\Gamma \left(\frac{n}{2}\right)}$ , $x>0$ , $n$ = positive integer and degrees of freedom

Formula

## F distribution

$X~{F}_{\mathrm{df}\left(n\right),\mathrm{df}\left(d\right)}$

$\mathrm{df}\left(n\right)=$ degrees of freedom for the numerator

$\mathrm{df}\left(d\right)=$ degrees of freedom for the denominator

$f\left(x\right)=\frac{\Gamma \left(\frac{u+v}{2}\right)}{\Gamma \left(\frac{u}{2}\right)\Gamma \left(\frac{v}{2}\right)}{\left(\frac{u}{v}\right)}^{\frac{u}{2}}{x}^{\left(\frac{u}{2}-1\right)}\left[1+\left(\frac{u}{v}\right){x}^{-.5\left(u+v\right)}\right]$

$X=\frac{{Y}_{u}}{{W}_{v}}$ , $Y$ , $W$ are chi-square

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