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4-9 EXPONENTIAL DISTRIBUTION 123

Therefore,

F1x2 P1X x2 1 e x , x 0
is the cumulative distribution function of X. By differentiating F(x), the probability density
function of X is calculated to be

f 1x2 e x , x 0

The derivation of the distribution of X depends only on the assumption that the flaws in
the wire follow a Poisson process. Also, the starting point for measuring X doesn’t matter
because the probability of the number of flaws in an interval of a Poisson process depends
only on the length of the interval, not on the location. For any Poisson process, the following
general result applies.

Definition
The random variable X that equals the distance between successive counts of a
Poisson process with mean 0 is an exponential random variable with parame-
ter . The probability density function of X is
x
f 1x2 e for 0 x (4-14)

The exponential distribution obtains its name from the exponential function in the proba-

bility density function. Plots of the exponential distribution for selected values of are shown

in Fig. 4-22. For any value of , the exponential distribution is quite skewed. The following
results are easily obtained and are left as an exercise.

If the random variable X has an exponential distribution with parameter ,

1 1
2
E1X2 and V1X2 (4-15)
2

It is important to use consistent units in the calculation of probabilities, means, and variances
involving exponential random variables. The following example illustrates unit conversions.
EXAMPLE 4-21 In a large corporate computer network, user log-ons to the system can be modeled as a Pois-
son process with a mean of 25 log-ons per hour. What is the probability that there are no log-
ons in an interval of 6 minutes?
Let X denote the time in hours from the start of the interval until the first log-on. Then, X
has an exponential distribution with 25 log-ons per hour. We are interested in the proba-
bility that X exceeds 6 minutes. Because is given in log-ons per hour, we express all time

units in hours. That is, 6 minutes 0.1 hour. The probability requested is shown as the shaded
area under the probability density function in Fig. 4-23. Therefore,

25x 2510.12
P1X 0.12 25e dx e 0.082
0.1