Introduction to Probability and Statistics for Engineers and Scientists

604 Chapter 14*:Life Testing

Method 2:This method uses the fact that

E[−log(1−F(X(i))]=

1

n

+

1

n− 1

+

1

n− 2

+ ···+

1

n−i+ 1

(14.5.7)

and then approximates−log(1−F(x(i))) by the foregoing. Thus, this second method
calls for setting

yi=log

[

1

n

+

1

(n−1)

+ ···+

1

(n−i+1)

]

(14.5.8)

REMARKS

(a) It is not, at present, clear which method provides superior estimates of the param-

eters of the Weibull distribution, and extensive simulation studies will be necessary

to determine this.

(b)Proofs of equalities 14.5.5 and 14.5.7 [which hold wheneverX(i)is theith small-

est of a sample of sizenfrom any continuous distributionF] are outlined in

Problems 28–30.

Problems..........................................................

1. A random variable whose distribution function is given by

F(t)= 1 −exp{−αtβ}, t≥ 0

is said to have a Weibull distribution with parametersα,β. Compute its failure

rate function.

2. IfXandYare independent random variables having failure rate functionsλx(t)
   andλy(t), show that the failure rate function ofZ=min(X,Y)is

λz(t)=λx(t)+λy(t)

3. The lung cancer rate of at-year-old male smoker,λ(t), is such that

λ(t)=.027+.025

(

t− 40

10

) 4

, t≥ 40

Assuming that a 40-year-old male smoker survives all other hazards, what is

the probability that he survives to(a)age 50,(b)age 60, without contracting

lung cancer? In the foregoing we are assuming that he remains a smoker throughout

his life.