Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

186 Some Special Distributions

(b)Ifr(x)=cebx,wherecandbare positive constants, show thatX has a

Gompertzcdf given by

F(x)=

{

1 −exp

{c

b(1−e

bx)} 0 <x<∞

0elsewhere.

(3.3.13)

This is frequently used by actuaries as a distribution of the length of human

life.

(c)Ifr(x)=bx, linear hazard rate, show that the pdf ofXis

f(x)=

{

bxe−bx

(^2) / 2
0 <x<∞
0elsewhere.
(3.3.14)
This pdf is called theRayleighpdf.
3.3.27.Write an R function that returns the valuef(x) for a specifiedxwhen
f(x) is the Weibull pdf given in expression (3.3.12). Next write an R function that
returns the associated hazard functionr(x). Obtain side-by-side plots of the pdf
and hazard function for the three cases:c=5andb=0.5;c=5andb=1.0; and
c=5andb=1.5.
3.3.28.In Example 3.3.5, a page of plots ofβpdfs was discussed. All of these pdfs
are mound shaped. Obtain a page of plots for all combinations ofαandβdrawn
from the set{. 25 ,. 75 , 1 , 1. 25 }. Comment on these shapes.
3.3.29.LetY 1 ,...,Ykhave a Dirichlet distribution with parametersα 1 ,...,αk,αk+1.
(a)Show thatY 1 has a beta distribution with parametersα=α 1 andβ=α 2 +
···+αk+1.
(b)Show thatY 1 +···+Yr,r≤k, has a beta distribution with parameters
α=α 1 +···+αrandβ=αr+1+···+αk+1.
(c)Show thatY 1 +Y 2 ,Y 3 +Y 4 ,Y 5 ,...,Yk,k≥5, have a Dirichlet distribution
with parametersα 1 +α 2 ,α 3 +α 4 ,α 5 ,...,αk,αk+1.
Hint: Recall the definition ofYiin Example 3.3.6 and use the fact that the
sum of several independent gamma variables withβ= 1 is a gamma variable.

3.4 TheNormalDistribution.........................

Motivation for the normal distribution is found in the Central Limit Theorem, which
is presented in Section 5.3. This theorem shows that normal distributions provide
an important family of distributions for applications and for statistical inference,
in general. We proceed by first introducing the standard normal distribution and
through it the general normal distribution.
Consider the integral

I=

∫∞

−∞

1

√

2 π

exp

(

−z^2

2

)

dz. (3.4.1)