Beta Distribution
Slide 15 of 47
The beta distribution is described by 2 parameters, € > 0 and € > 0. The mean of the distribution is given by €/(€ + €) and the variance as €€/[(€€+€€)€2€(€€+€€€+€1)], with the mode (€ - 1)/(€ + € - 2) (mode only for € € 1). Most random number generation techniques for the beta distribution require you to specify values for € and €. For a given mean (µ) and variance (€€2€) or standard deviation (€),
stackalign{
alpha ~=&~ {mu sup 2 (1~-~mu)} over {sigma sup 2} ~-~ mu , ~
func { a n d } #
beta ~=&~ {left ( sigma sup 2 ~+~ mu (mu ~-~ 1) right )
(mu ~-~ 1)} over { sigma sup 2 } ~. }
However, the amount of variation possible is limited because the distribution is bounded on the [0,1] interval. Thus, for a mean of 0.5, the maximum variance approaches 0.25 as € and € approach zero.
