Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

176 Some Special Distributions

0 5 10 15

0.00

0.15

0.30

x

f(x

)

α = .25

α = .5

α = 1

β = 4

0 5 10 15 20 25 30 35

0.00

0.06

0.12

x

f(x)

α = 4

β = 2

β = 3

β = 4

Figure 3.3.1: Several gamma densities

Note that the hazard functionr(x)satisfies−(d/dx)log[1−F(x)]; that is,

1 −F(x)=e−

R

r(x)dx+c, (3.3.4)

for a constantc. When the support ofXis (0,∞),F(0) = 0 serves as a boundary

condition to solve forc. In practice, often the scientist can describe the hazard rate

and, hence,F(x) can be determined from expression (3.3.4). For example, suppose

the hazard rate ofXis constant; i.e,r(x)=1/βfor someβ>0. Then

1 −F(x)=e−

R(1/β)dx+c

=e−x/βec.

SinceF(0) = 0,ec=1. SothepdfofXis

f(x)=

{ 1

βe

−x/β x> 0

0elsewhere.

(3.3.5)

Of course, this is a Γ(1,β) distribution, but it is also called theexponential dis-

tributionwith parameter 1/β. An important property of this distribution is given

in Exercise 3.3.25.

Using R, hazard functions can be quickly plotted. Here is the code for an

overlay plot of the hazard functions of the exponential distribution withβ=8and

the Γ(4,2)-distribution.

x=seq(.1,15,.1); t=dgamma(x,shape=4,scale=2)

b=(1-pgamma(x,shape=4,scale=2));y1=t/b;plot(y1~x);abline(h=1/8)

Note that the hazard function of this Γ-distribution is an increasing function of

x; i.e., the rate of failure increases as time progresses. Other examples of hazard

functions are given in Exercise 3.3.26.