Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.7.∗Mixture Distributions 223

EXERCISES

3.7.1.SupposeY has a Γ(α, β) distribution. LetX=eY. Show that the pdf of
Xis given by expression (3.7.5). Determine the cdf ofXintermsofthecdfofa
Γ-distribution. Derive the mean and variance ofX.

3.7.2.Write R functions for the pdf and cdf of the random variable in Exercise

3.7.1.

3.7.3.In Example 3.7.1, derive the pdf of the mixture distribution given in expres-

sion (3.7.6), then obtain its mean and variance as given in expressions (3.7.7) and

(3.7.8).

3.7.4.Using the R function for the pdf in Exercise 3.7.2 anddgamma,writeanR

function for the mixture pdf (3.7.6). Forα=β= 2, obtain a page of plots of this

density forp=0. 05 , 0. 10 , 0 .15 and 0.20.

3.7.5.Consider the mixture distribution (9/10)N(0,1) + (1/10)N(0,9). Show that

its kurtosis is 8.34.

3.7.6.LetXhave the conditional geometric pmfθ(1−θ)x−^1 ,x=1, 2 ,...,whereθ
is a value of a random variable having a beta pdf with parametersαandβ. Show
that the marginal (unconditional) pmf ofXis

Γ(α+β)Γ(α+1)Γ(β+x−1)

Γ(α)Γ(β)Γ(α+β+x)

,x=1, 2 ,....

Ifα=1,weobtain
β
(β+x)(β+x−1)

,x=1, 2 ,...,

which is one form ofZipf’s law.

3.7.7. Repeat Exercise 3.7.6, lettingXhave a conditional negative binomial dis-
tribution instead of the geometric one.

3.7.8.LetXhave a generalized Pareto distribution with parametersk,α,andβ.
Show, by change of variables, thatY=βX/(1 +βX) has a beta distribution.

3.7.9.Show that the failure rate (hazard function) of the Pareto distribution is

h(x)

1 −H(x)

=

α

β−^1 +x

.

Find the failure rate (hazard function) of the Burr distribution with cdf

G(y)=1−

(

1

1+βyτ

)α

, 0 ≤y<∞.

In each of these two failure rates, note what happens as the value of the variable

increases.