Probabilities for the exponential probability distribution.

For the exponential probability distribution with the mean number of occurrences per unit of time equal to ,

=

.

Example:

A processing machine breaks down an average of once in four weeks. What is the probability that the next breakdown will not occur for at least six weeks after the previous breakdown? Assume that the time between breakdowns has an exponential distribution.

Solution:

Let x denote the lapse time between any two successive breakdowns of this machine. We are to find the probability

Because the unit of time for x is in the weeks, we must define the mean number of breakdowns per week. Since there is one breakdown in four weeks,

.

The required probability is calculated using the formula = .

In our example, and . The required probability is

= .

The value of can be found from the Table 1 of the Appendix.

Example:

A teller at the bank serves, on average, 30 customers per hour. Assume that the service time for a customer has an exponential distribution.

a) What is the probability that the next customer will take five minutes or more to be served?

b) Find the probability that the next customer will take two minutes or less to be served.

c) What is the probability the next customer will take two to four minutes to be served?

Solution:

Let x be the time taken by this teller to serve a customer. We must find the mean number of customers served per minute by this teller to define per unit of time (minute). The teller serves on average 30 customers per 60 minutes. Hence

=30/60=0.5 customers served per minute.

a) We need to find the probability . (Fig. 4.23.)

In this case a =5.

= =

The probability is 0.0821 that a customer will take more than five minutes

to be served.

b) We are to find . This probability will be calculated using the formula = . In this case a = 2 minutes.

So

=

.

Thus, the probability is 0.6321 that a customer will be served in two minutes or less.

c)

Thus, the probability that the teller will take two to four minutes to serve a customer is 0.2325.

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