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.