Beta

Parameters

2 (called \(\alpha \) and \(\beta \) below)

Kernel

\(x^{\alpha -1}\left( 1-x\right) ^{\beta -1}\)

Support

\(\left[ 0,1\right] \) . If \(\alpha >1\), the density is 0 at \(x=0\) ; if \(\beta >1\), the density is 0 at \(x=1\)

Mean

\(\dfrac{\alpha }{\alpha +\beta }\)

Variance

\(\dfrac{\alpha \beta }{\left( \alpha +\beta \right)^{2}(\alpha +\beta +1)}\)

Main Uses

Priors and approximate posteriors for parameters that measure fractions, or probabilities, or autoregressive coefficients (when a negative value is unreasonable).

Density Function

%LOGBETADENSITY(x,a,b)

Random Draws

%RANBETA(a,b) draws one or more (depending on the target) independent Betas

Moment Matching

%BetaParms(mean,sd) (external function) returns the 2-vector of parameters for a beta with the given mean and standard deviation.

Extensions

If \(x\sim Beta(\alpha ,\beta )\) then \(ax+b\) is distributed on \(\left[ a,a+b\right] \)