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Beta Distribution
Statistics - Beta Distribution
The beta distribution represents continuous probabipty distribution parametrized by two positive shape parameters, $ alpha $ and $ eta $, which appear as exponents of the random variable x and control the shape of the distribution.
Probabipty density function
Probabipty density function of Beta distribution is given as:
Formula
${ f(x) = frac{(x-a)^{alpha-1}(b-x)^{eta-1}}{B(alpha,eta) (b-a)^{alpha+eta-1}} hspace{.3in} a le x le b; alpha, eta > 0 \[7pt] , where B(alpha,eta) = int_{0}^{1} {t^{alpha-1}(1-t)^{eta-1}dt} }$Where −
${ alpha, eta }$ = shape parameters.
${a, b}$ = upper and lower bounds.
${B(alpha,eta)}$ = Beta function.
Standard Beta Distribution
In case of having upper and lower bounds as 1 and 0, beta distribution is called the standard beta distribution. It is driven by following formula:
Formula
${ f(x) = frac{x^{alpha-1}(1-x)^{eta-1}}{B(alpha,eta)} hspace{.3in} le x le 1; alpha, eta > 0}$Cumulative distribution function
Cumulative distribution function of Beta distribution is given as:
Formula
${ F(x) = I_{x}(alpha,eta) = frac{int_{0}^{x}{t^{alpha-1}(1-t)^{eta-1}dt}}{B(alpha,eta)} hspace{.2in} 0 le x le 1; p, eta > 0 }$Where −
${ alpha, eta }$ = shape parameters.
${a, b}$ = upper and lower bounds.
${B(alpha,eta)}$ = Beta function.
It is also called incomplete beta function ratio.
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