Fundamentals of Probability and Statistics for Engineers

increasing portion of the h(t) curve. System reliability can be optimized by
in itial ‘burn-in’until time t 1 to avoid premature failure and by part replacement
at time t 2 to avoid wear out.
We can now show that the exponential failure law is appropriate during the
‘in-usage’ period of a system’s normal life. Substituting

and

into Equation (7.65), we immediately have

We see from the above that parameter in the exponential distribution plays
the role of a (constant) failure rate.
We have seen in Example 7.7 that the gamma distribution is appropriate
to describe the time required for a total of arrivals. In the context of
failure laws, the gamma distribution can be thought of as a generalization of
the exponential failure law for systems that fail as soon as exactly events
fail, assuming events take place according to the Poisson law. Thus, the
gamma distribution is appropriate as a time-to-failure model for systems
having one operating unit and 1 standby units; these standby units go
into operation sequentially, and each one has an exponential time-to-failure
distribution.

7.4.2 Chi-Squared Distribution

Another important special case of the gamma distribution is the chi-squared
(^2 ) distribution, obtained by setting 1/2 and n/2 in Equation (7.52),
where n is a positive integer. The^2 distribution thus contains one parameter,
n, with pdf of the form

Some Important Continuous Distributions 219

fT…t†ˆet;t 0 ;

FT…t†ˆ 1 et;t 0 ;

h…t†ˆ: … 7 : 66 †









 

ˆˆ

fX…x†ˆ

1

2 n=^2 …n= 2 †

x…n=^2 †^1 ex=^2 ; forx 0 ;

0 ; elsewhere:

8

><

>:

… 7 : 67 †