Fundamentals of Probability and Statistics for Engineers

7.4.1 Exponential Distribution

When 1, the gamma density function given by Equation (7.52) reduces to
the exponential form

where is the parameter of the distribution. Its associated PDF, mean,
and variance are obtained from Equations (7.55) and (7.57) by setting 1.
They are

and

Among many of its applications, two broad classes stand out. First, we will
show that the exponential distribution describes interarrival time when arrivals
obey the Poisson distribution. It also plays a central role in reliability, where the
exponential distribution is one of the most important failure laws.

7.4.1.1 Interarrival Time

There is a very close tie between the Poisson and exponential distributions. Let
random variable X(0,t) be the number of arrivals in the time interval [0,t) and
assume that it is Poisson distributed. Our interest now is in the time between
two successive arrivals, which is, of course, also a random variable. Let this
interarrival time be denoted by T. Its probability distribution function, FT (t),
is, by definition,

In terms of X(0,t), the event T > t is equivalent to the event that there
are no arrivals during time interval [0, t), or X(0, t) 0. Hence, since

Some Important Continuous Distributions 215

ˆ

fX…x†ˆ

ex; forx 0 ;

0 ; elsewhere;



… 7 : 58 †

">0)

ˆ

FX…x†ˆ

ˆ 1 ex; forx 0 ;

ˆ 0 ; elsewhere;



… 7 : 59 †

mXˆ

1



;^2 Xˆ

1

^2

: … 7 : 60 †

FT…t†ˆ

P…Tt†ˆ 1 P…T>t†; fort 0 ;

0 ; elsewhere:



… 7 : 61 †

ˆ