Page 232 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 215
7.4.1 EXPONENTIAL DISTRIBUTION

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

x
e ; for x 0;
f x†ˆ 7:58†
X
0; elsewhere;
where " > 0) is the parameter of the distribution. Its associated PDF, mean,
and variance are obtained from Equations (7.55) and (7.57) by setting ˆ 1.
They are

x
ˆ 1 e ; for x 0;
F X x†ˆ 7:59†
ˆ 0; elsewhere;

and
1 2 1
m X ˆ ; ˆ : 7:60†
X
2
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, F T (t),
is, by definition,

P T t†ˆ 1 P T > t†; for t 0;
F T t†ˆ 7:61†
0; elsewhere:

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) ˆ 0. Hence, since

TLFeBOOK