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 Poisson probability 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

                       

                                   

 

Exercises

1. Using the Poisson formula, find the following probabilities

       a)  for

       b)  for

2. Let x be a Poisson random variable. Using the Poisson probabilities table, write the probability distribution of x for each of the following. Find the mean and standard deviation for each of these probability distributions.

       a) ;              b)

3. An average of 7.5 crimes are reported per day to police in a city. Use the Poisson formula to find the probability that

a) exactly 3 crimes will be reported to a police on a certain day

b) at least 2 crimes will be reported to a police on a certain day.

4. A mail-order company receives an average 1.3 complaints per day. Find the probability that it will receive

a) exactly 3 complaints

b) 2 to 3 complaints

c) more than 3 complaints

d) less than 3 complaints on a certain day.

5. An average of 4.5 customers come to the bank per half hour.

a) Find the probability that exactly 2 customers will come to this bank during a given hour;

b) Find the probability that during a given hour, the number of customers who will come to the bank is at most 2.

6. An average of 0.6 accidents occur per month at a large company.

a) Find the probability that no accident will occur at this company during a given month.

b) Find the mean, variance, and standard deviation of the number of accidents that will occur at this company during a given month.

 

                          Answers

1. a) 0.1991; b) 0.0771; 2. a) ;  b) ;

3. a) 0.03888; b) 0.9953; 4. a) 0.0998; b) 0.3301; c) 0.0431; d) 0.8569;

5. a) 0.0050; b) 0.0062; 6. a) 0.5488; b) .

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