6.4.2. The Poisson Probability Distribution

Properties of Poisson Experiment:

1. X : representing the number of occurrences in a continuous interval.

expected value of occurrences in this interval.

2. The probability of an occurrence is the same for any two intervals of equal length!!

The expected value of occurrences in an interval is proportional to the length of this interval.

3. The occurrence or nonoccurrence in any interval is independent of the occurrence or

nonoccurrence in any other interval.

4. The probability of two or more occurrences in a very small interval is close to 0.

Poisson Probability Distribution:

Let X be the random variable representing the number of occurrences of a Poisson experiment

in some interval. Then, the probability distribution function for X is

,

where and is some parameter.

Properties of Poisson Probability Distribution:

A random variable X has the Poisson probability distribution with parameter ,

then

and

.

Example 5:

Suppose the average number of car accidents on the highway in one day is 4. What is the

probability of no car accident in one day? What is the probability of 1 car accidence in two

days?

[solution:]

It is sensible to use Poisson random variable representing the number of car accidents on the

high way. Let X representing the number of car accidents on the high way in one day. Then,

and

.

Then,

Since the average number of car accidents in one day is 4, thus the average number of car

accidents in two days should be 8. Let Y represent the number of car accidents in two days.

Then,

and

.

Then,

Example 6:

Suppose the average number of calls by 104 in one minute is 2. What is the probability of

10 calls in 5 minutes?

[solution]:

Since the average number of calls by 104 in one minute is 2, thus the average number of

calls in 5 minutes is 10. Let X represent the number of calls in 5 minutes. Then,

and

.

Then,

.

JavaStatSoft:

Probability density:

Statistics -> Probability -> Probability Functions -> Probability Density Function (PDF)

Probability density:

Statistics -> Probability -> Probability Functions -> Cumulative Distribution Function (CDF)

Percentile (or Quantile):

Statistics -> Probability -> Probability Functions -> Percentile Function