3.3. TheΓ,χ^2 ,andβDistributions 181In accordance with Theorem 2.4.1 the random variables are independent. The
marginal pdf ofY 2 isg 2 (y 2 )=y 2 α−^1 (1−y 2 )β−^1
Γ(α)Γ(β)∫∞0yα 1 +β−^1 e−y^1 dy 1={
Γ(α+β)
Γ(α)Γ(β)yα− 1
2 (1−y^2 )β− (^10) <y 2 < 1
0elsewhere.
(3.3.9)
This pdf is that of thebeta distributionwith parametersαandβ.Sinceg(y 1 ,y 2 )≡
g 1 (y 1 )g 2 (y 2 ), it must be that the pdf ofY 1 is
g 1 (y 1 )=
{
1
Γ(α+β)y
α+β− 1
1 e
−y (^10) <y 1 <∞
0elsewhere,
which is that of a gamma distribution with parameter values ofα+βand 1.
It is an easy exercise to show that the mean and the variance ofY 2 , which has
a beta distribution with parametersαandβ, are, respectively,
μ=
α
α+β
,σ^2 =
αβ
(α+β+1)(α+β)^2
.
The package R calculates probabilities for the beta distribution. IfXhas a beta
distribution with parametersα=aandβ=b, then the commandpbeta(x,a,b)
returnsP(X≤x) and the commanddbeta(x,a,b)returns the value of the pdf of
Xatx.
Example 3.3.5(Shapes ofβ-Distributions).The following 3 lines of R code^5 , will
obtain a 4×4 page of plots ofβpdfs for all combinations of integer values ofαand
βbetween 2 and 5. Those distributions on the main diagonal of the page of plots
are symmetric, those below the main diagonal are left-skewed, and those above the
main diagonal are right-skewed.
par(mfrow=c(4,4));r1=2:5; r2=2:5;x=seq(.01,.99,.01)
for(a in r1){for(b in r2){plot(dbeta(x,a,b)~x);
title(paste("alpha = ",a,"beta = ",b))}}
Note that ifα=β= 1, then theβ-distribution is the uniform distribution with
support (0,1).
We close this section with another example of a random variable whose distri-
bution is derived from a transformation of gamma random variables.
Example 3.3.6(Dirichlet Distribution).LetX 1 ,X 2 ,...,Xk+1be independent ran-
dom variables, each having a gamma distribution withβ= 1. The joint pdf of these
variables may be written as
h(x 1 ,x 2 ,...,xk+1)=
{∏
k+1
i=1
1
Γ(αi)x
αi− 1
i e
−xi 0 <xi<∞
0elsewhere.
(^5) Download the R function betaplts at the site listed in the Preface.