The betadistribution is aprobabilitydistributionover(0,1)thatisoftenusedinapplications for whichtherandomvariableisaproportion.Thebeta pdf is for parameters and , where () denotes thegammafunction. (a) Showthattheuniformdistributionisthespecialcase = = 1. (b)

The betadistribution is aprobabilitydistributionover(0,1)thatisoftenusedinapplications for whichtherandomvariableisaproportion.Thebeta pdf is

for parameters α and β, where Γ(⋅) denotes thegammafunction.

(a) Showthattheuniformdistributionisthespecialcase α = β = 1.

(b) Showthat μ = E(Y ) = α~(α + β).

(c) Find E(Y 2). Showthatvar(Y ) = αβ~(α + β)2(α + β + 1) = μ(1 − μ)~(α + β + 1). Forfixed α + β, notethatvar(Y ) decreases as μ approaches0or1.

(d) Usingafunctionsuchas dbeta in R, plotthebeta pdf for (i) α = β = 0.5, 1.0,10,100,(ii)
some valuesof α > β and somevaluesof α

f(y; a, )= T(a+B) (a)(3) -ya-1 (1-y)-1, 0y1, (2.20)