Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

224 Some Special Distributions

3.7.10.For the Burr distribution, show that

E(Xk)=

1

βk/τ

Γ

(

α−

k

τ

)

Γ

(

k

τ

+1

)/

Γ(α),

providedk<ατ.

3.7.11.Let the numberXof accidents have a Poisson distribution with meanλθ.
Supposeλ, the liability to have an accident, has, givenθ, a gamma pdf with pa-
rametersα=handβ=h−^1 ;andθ, an accident proneness factor, has a generalized
Pareto pdf with parametersα,λ=h,andk. Show that the unconditional pdf of
Xis

Γ(α+k)Γ(α+h)Γ(α+h+k)Γ(h+k)Γ(k+x)

Γ(α)Γ(α+k+h)Γ(h)Γ(k)Γ(α+h+k+x)x!

,x=0, 1 , 2 ,...,

sometimes called thegeneralized Waringpmf.

3.7.12.LetXhave a conditional Burr distribution with fixed parametersβandτ,
given parameterα.

(a)Ifαhas the geometric pmfp(1−p)α,α=0, 1 , 2 ,..., show that the uncondi-

tional distribution ofXis a Burr distribution.

(b)Ifαhas the exponential pdfβ−^1 e−α/β,α>0, find the unconditional pdf of

X.

3.7.13.LetXhave the conditional Weibull pdf

f(x|θ)=θτxτ−^1 e−θx

τ

, 0 <x<∞,

and let the pdf (weighting function)g(θ) be gamma with parametersαandβ. Show

that the compound (marginal) pdf ofXis that of Burr.

3.7.14.IfXhas a Pareto distribution with parametersαandβand ifcis a positive
constant, show thatY=cXhas a Pareto distribution with parametersαandβ/c.