14.2Hazard Rate Functions 583

which implies that

1 −F(t)=exp

{

−

∫t

0

λ(s)ds

}

(14.2.1)

Hence a distribution function of a positive continuous random variable can be specified
by giving its hazard rate function. For instance, if a random variable has a linear hazard
rate function — that is, if

λ(t)=a+bt

then its distribution function is given by

F(t)= 1 −e−at−bt

(^2) /2
and differentiation yields that its density is
f(t)=(a+bt)e−(at+bt
(^2) /2)
, t≥ 0
Whena=0, the foregoing is known as theRayleigh density function.
EXAMPLE 14.2a One often hears that the death rate of a person that smokes is, at each
age, twice that of a nonsmoker. What does this mean? Does it mean that a nonsmoker has
twice the probability of surviving a given number of years as does a smoker of the same
age?
SOLUTION Ifλs(t) denotes the hazard rate of a smoker of agetandλn(t) that of a
nonsmoker of aget, then the foregoing is equivalent to the statement that
λs(t)= 2 λn(t)
The probability that anA-year-old nonsmoker will survive until ageB,A<B,is
P{A-year-old nonsmoker reaches ageB}
=P{nonsmoker’s lifetime>B|nonsmoker’s lifetime>A}

1 −Fnon(B)
1 −Fnon(A)

exp
{
−
∫B
0 λn(t)dt
}
exp
{
−
∫A
0 λn(t)dt
} from Equation 14.2.1
=exp
{
−
∫B
A
λn(t)dt
}