Page 252 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 235

and, if it is continuous,
k z " k 1 z " k
f z†ˆ exp ; k > 0; w >"; z ": 7:121†
Z
w " w " w "
The mean and variance of Z are

9

1
m Z ˆ " ‡ w "† 1 ‡ ; >
>
k =
>
7:122†

2 2 1 >
2
2
ˆ w "† 1 ‡ 1 ‡ >
Z : >
k k
;
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
k 1
k t
h t†ˆ ; t 0; 7:123†
w w
is capable of assuming a wide variety of shapes, and its associated probability
density function for T, the time to failure, is given by
k 1
k t k
t
f t†ˆ exp‰ Š; w; k > 0; t 0: 7:124†
T
w w w
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 " ˆ 0.
The relationship between Type-III and Type-I minimum-value asymptotic
distributions can also be established. Let Z I and Z III be the random variables
having, respectively, Type-I and Type-III asymptotic distributions of minimum
values. Then

‰ln z "†Š; z "; 7:125†
F Z III z†ˆ F Z I
with u ˆ ln (w " ), and ˆ k. If they are continuous, the relationship between
their pdfs is

1
f z†ˆ f ‰ln z "†Š; z ": 7:126†
z "
Z III Z I

TLFeBOOK