The exponential probability distribution

The exponential probability distribution is another important probability density function. This probability distribution is closely related to the Poisson probability distribution.

The exponential probability distribution has only one parameter , which denotes the average number of occurrences per unit of time.

Remark:

The exponential distribution differs from the normal distribution in two important way

1. it is restricted to random variables with positive values;

2. its distribution is not symmetric.

Definition:

The exponential random variable X (x >0) has a probability density function

for

where is the mean number of occurrences per unit time, x is the number of time units until the next occurrence, and , then X is said to follow an exponential probability distribution. It can be shown that is the same parameter used for the Poisson distribution and that the mean time between occurrences is .

The cumulative distribution function is

for

The distribution has mean and variance .

The probability for the exponential probability distribution is given

by the area in the tail of the exponential probability distribution curve beyond , as it shown in Figure 4.22.

As we know from earlier discussion, for a continuous random variable x, is equal to . Hence for an exponential probability distribution,

= =

By using the complementary probability rule, we obtain:

The probability that the x is between two successive occurrences is in the interval “ a ” to “ b ” is

.

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