Introduction to Probability and Statistics for Engineers and Scientists

582 Chapter 14*:Life Testing

P{X∈(t,t+dt)|X>t}. Now

P{X∈(t,t+dt)|X>t}=

P{X∈(t,t+dt),X>t}

P{X>t}

=

P{X∈(t,t+dt)}

P{X>t}

≈

f(t)

1 −F(t)

dt

That is,λ(t) represents the conditional probability intensity that an item of agetwill fail
in the next moment.
Suppose now that the lifetime distribution is exponential. Then, by the memoryless
property of the exponential distribution it follows that the distribution of remaining life
for at-year-old item is the same as for a new item. Henceλ(t) should be constant, which
is verified as follows:

λ(t)=

f(t)

1 −F(t)

=

λe−λt

e−λt

=λ

Thus, the failure rate function for the exponential distribution is constant. The parameter
λis often referred to as therateof the distribution.
We now show that the failure rate functionλ(t),t ≥ 0, uniquely determines the
distributionF. To show this, note that by definition

λ(s)=

f(s)

1 −F(s)

=

d

dsF(s)

1 −F(s)

=

d

ds

{−log[ 1 −F(s)]}

Integrating both sides of this equation from 0 totyields

∫t

0

λ(s)ds=−log[ 1 −F(t)]+log[ 1 −F(0)]

=−log[ 1 −F(t)] sinceF(0)= 0