Fundamentals of Probability and Statistics for Engineers

and, if it is continuous,

The mean and variance of Z are

We have seen in Section 7.4.1 that the exponential distribution is used as a
failure law in reliability studies, which corresponds to a constant hazard func-
tion [see Equations (7.64) and (7.66)]. The distribution given by Equations
(7.120) and (7.121) is frequently used as a generalized time-to-failure model
for cases in which the hazard function varies with time. One can show that the
hazard function

is capable of assuming a wide variety of shapes, and its associated probability
density function for T, the time to failure, is given by

It is the so-called Weibull distribution, after Weibull, who first obtained it,
heuristically (Weibull, 1939). Clearly, Equation (7.124) is a special case of
Equation (7.121), with 0.
The relationship between Type-III and Type-I minimum-value asymptotic
distributions can also be established. Let ZI and ZIII be the random variables
having, respectively, Type-I and Type-III asymptotic distributions of minimum
values. Then

with u ln (w ), and k. If they are continuous, the relationship between
their pdfs is

Some Important Continuous Distributions 235

fZ…z†ˆ

k

w"

z"

w"

k 1

exp

z"

w"

k

; k> 0 ;w>";z": … 7 : 121 †

mZˆ"‡…w"† 1 ‡

1

k



;

^2 Zˆ…w"†^2  1 ‡

2

k



^21 ‡

1

k



:

9

>>

>=

>>

>;

… 7 : 122 †

h…t†ˆ

k

w

t

w

k 1

; t 0 ; … 7 : 123 †

fT…t†ˆ

k

w

t

w

k 1

exp‰

t

w

k

Š; w;k> 0 ;t 0 : … 7 : 124 †

"ˆ

FZIII…z†ˆFZI‰ln…z"†Š; z"; … 7 : 125 †

ˆ" ˆ

fZIII…z†ˆ

1

z"

fZI‰ln…z"†Š; z": … 7 : 126 †