The Poisson probability distribution

The Poisson probability distribution, named after the French mathematician Siemon D. Poisson, is another important probability distribution of a discrete random variable that has a large number of applications.

A Poisson probability distribution is modeled according to certain assumptions:

1. x is a discrete random variable;

2. The occurrences are random.

3. The occurrences are independent.

In the Poisson probability distribution terminology, the average number of occurrences in an interval is denoted by (Greek letter lambda). The actual number of occurrences in that interval is denoted by x.

Poisson probability distribution formula:

According to the Poisson probability distribution, the probability of x occurrences in an interval is

where

the probability of x successes over an interval;

is the mean number of occurrences in that interval

(the base of natural logarithms)

The mean and variance of the Poissonprobability distribution are:

and .

Remark: As it is obvious from the Poisson probability distribution formula, we need to know only the value to compute the probability of any given value of x. We can read the value of for a given from Table1 of the Appendix.

Example:

A computer breaks down at an average of three times per month. Using the Poisson probability distribution formula, find the probability that during the next month this computer will have

a) exactly three breakdowns;

b) at most one breakdown.

Solution:

Let be the mean number of breakdowns per month and x be the actual number of breakdowns observed during the next month for this computer. Then =3.

a) The probability that exactly three breakdowns will be observed during the next month is

b) The probability that at most one breakdown will be observed during the next month is given by the sum of the probabilities of zero and one breakdown. Then

.

Example:

A car salesperson sells an average of 0.9 cars per day. Find the probability of selling

a) exactly 2

b) at least 3 cars per day

c) find the mean, variance and standard deviation of selling cars per day.

Solution:

Let be the mean number of cars sold per day by this salesperson.

Let x be the number of cars sold by this salesperson. Hence, =0.9

a) .

b)

.

c)

and

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