Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

182 Some Special Distributions

Let

Yi=

Xi

X 1 +X 2 +···+Xk+1

,i=1, 2 ,...,k,

andYk+1=X 1 +X 2 +···+Xk+1denotek+1 new random variables. The associated

transformation mapsA={(x 1 ,...,xk+1):0<xi<∞,i=1,...,k+1}onto the

space:

B={(y 1 ,...,yk,yk+1):0<yi,i=1,...,k, y 1 +···+yk< 1 , 0 <yk+1<∞}.

The single-valued inverse functions arex 1 = y 1 yk+1,...,xk =ykyk+1,xk+1 =
yk+1(1−y 1 −···−yk), so that the Jacobian is

J=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

yk+1 0 ··· 0 y 1

0 yk+1 ··· 0 y 2

..

.

..

.

..

.

..

.

00 ··· yk+1 yk

−yk+1 −yk+1 ··· −yk+1 (1−y 1 −···−yk)

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

=ykk+1.

Hence the joint pdf ofY 1 ,...,Yk,Yk+1is given by

yαk+1^1 +···+αk+1−^1 yα 11 −^1 ···yαkk−^1 (1−y 1 −···−yk)αk+1−^1 e−yk+1

Γ(α 1 )···Γ(αk)Γ(αk+1)

,

provided that (y 1 ,...,yk,yk+1)∈Band is equal to zero elsewhere. By integrating

outyk+1, the joint pdf ofY 1 ,...,Ykis seen to be

g(y 1 ,...,yk)=

Γ(α 1 +···+αk+1)

Γ(α 1 )···Γ(αk+1)

y 1 α^1 −^1 ···ykαk−^1 (1−y 1 −···−yk)αk+1−^1 ,(3.3.10)

when 0<yi,i=1,...,k, y 1 +···+yk<1, while the functiongis equal to zero
elsewhere. Random variablesY 1 ,...,Ykthat have a joint pdf of this form are said to
have aDirichlet pdf. It is seen, in the special case ofk= 1, that the Dirichlet pdf
becomes a beta pdf. Moreover, it is also clear from the joint pdf ofY 1 ,...,Yk,Yk+1
thatYk+1has a gamma distribution with parametersα 1 +···+αk+αk+1andβ=1
and thatYk+1is independent ofY 1 ,Y 2 ,...,Yk.

EXERCISES

3.3.1.Suppose (1− 2 t)−^6 ,t<^12 is the mgf of the random variableX.

(a)Use R to computeP(X< 5 .23).

(b)Find the meanμand varianceσ^2 ofX. Use R to computeP(|X−μ|< 2 σ).

3.3.2.IfXisχ^2 (5), determine the constantscanddso thatP(c<X<d)=0. 95
andP(X<c)=0.025.