Beta Distribution

A continuous Random Variable with range $[0,1]$ is of beta distribution if

• Parameters
  • $a,b>0$
• PDF
  • $f(x)=\frac{\Gamma (a+b)}{\Gamma (a)\Gamma (b)}{x}^{a-1}(1-x{)}^{b-1}$
  • where $\Gamma$ is the Gamma Function
• Mean
  • $\frac{a}{a+b}$
• Variance
  • $\frac{ab}{(a+b{)}^2(a+b+1)}$
• MGF
  • $\frac{\Gamma (a+b)}{\Gamma (a)\Gamma (b)}{\int }_0^1{x}^{a-1}(1-x{)}^{b-1}{e}^{tx}\mathrm{d}x$

Properties

• Uniform Distribution is a special case of beta distribution with $a=b=1$.
• Beta-Binomial model:Given a prior $\mathrm{Beta}(a,b)$, the posterior distribution of $p$ given $n$ Bernoulli trials with $k$ successes is $\mathrm{Beta}(a+k,b+n-k)$.
• Uniform Distribution: If $X\sim \mathrm{U}(0,1)$ and $a>0$ then $X^{1/a}\sim \mathrm{Beta}(a,1)$.
• The $k$-th order statistic of a sample of size $n$ from the Uniform Distribution is a beta random variable, $U_{(k)}\sim \mathrm{Beta}(k,n+1-k)$; see also ^2e635f.
• $\mathrm{Beta}(\alpha n,\beta n)\to \mathcal{N}\left(\frac{\alpha }{\alpha +\beta },\frac{\alpha \beta }{(\alpha +\beta {)}^{3}n}\right)$
• Gamma Distribution: If $X\sim \mathrm{Gamma}(\alpha ,\theta )$ and $Y\sim \mathrm{Gamma}(\beta ,\theta )$ are independent, then $\frac{X}{X+Y}\sim \mathrm{Beta}(\alpha ,\beta )$.
• $n\mathrm{Beta}(k,n)\to \mathrm{Gamma}(k,1)$, $n\mathrm{Beta}(1,n)\to \mathrm{Exp}(1)$
• Chi-Square Distribution: If $X\sim {\chi }^2(\alpha )$ and $Y\sim {\chi }^2(\beta )$ are independent, then $\frac{X}{X+Y}\sim \mathrm{Beta}\left(\frac{\alpha }{2},\frac{\beta }{2}\right)$.
• Cauchy Distribution: If $X\sim \mathrm{Cauchy}(0,1)$ then $\frac{1}{1+X^2}\sim \mathrm{Beta}\left(\frac{1}{2},\frac{1}{2}\right)$