The Poisson Probability Distribution

If In a binomial experiment you are interested in finding the probability of a specific number of occurrences takes place within a given specified interval (usually time or space), you can use the Poisson distribution. For instance, the probability that an employee will take 15 sick days within a year, you can use the Poisson distribution.

Definition:

The Poisson distribution is a discrete probability distribution of a random variable x that satisfies the following conditions.

1. The experiment consists of counting the number of times, x, an event occurs in a given interval. The interval can be an interval of time, area, or volume.
2. The probability of two or more success in any sufficiently small subinterval is 0. For example, the fixed interval might be any time between 0 and 5 minutes. A subinterval could be any time between 1 and 2 minutes.
3. The probability of the event occurring is the same for any two intervals of equal length.
4. The number of occurrences (success) in any interval is independent of the number of occurrences in any other interval provided the intervals are not overlapping.

The probability of exactly x occurrences in an interval is

Where e is an irrational number approximately equal to 2.71828 and is the mean number of occurrences per interval unit.

Example 1:

The mean number of accidents per month at a certain intersection is three.

a)What is the probability that in any given month four accidents will occur at this intersection?

b)What is the probability that more than four accidents will occur in any given month at the intersection?

In Example 1 you used a formula to determine a Poisson probability. You can also use a table to find Poisson probabilities. Table 4 lists the Poisson probability for selected values of x and . You can also use technology tools, such as MINITAB, Excel, and the TI-83/84, to find the Poisson probabilities as well.

Finding Poisson Probabilities Using a Table and TI83-84:

Example 2:

A population count shows that there is an average of 3.6 rabbits per acre living in a field. Use a table to find the probability that two rabbits are found on any given acre for the field.

1st method: Using Table

2nd method: Using TI83-84

Step1: press 2nd VARS to access the probability distribution menu.

Step2: Highlight poissonpdef and hit enter.

Step3; With poissonpdf ( on the HOME screen, type the value of mean, followed by the number of successes, x.

Finding the Mean and Standard Deviation of a Poisson Random Variable:

If cars arrive at McDonald’s at the rate of 2 per minute between 12:00 noon and 1:00 p.m., how many cars would you expect to arrive between the hours of 12:00 noon and 12:05 p.m.? Considering that 2 cars arrive every minute (on average) and we are observing the arrival of cars for 5 minutes, it might seem reasonable that we expect 2(5)=10 cars to arrive. Since the expected value of a random variables is the mean of the random variable, it seems reasonable that for interval t. (the Greek letter Lambda) represents the average number of occurrences of the event in some interval of length 1.

A random variable x that follows a Poisson process with parameter has mean (or expected value) and standard deviation given by the formulas

and σ = = where t is the length of the interval.

Example 3:

A biologist performs an experiment in which 2000 Asian beetles are allowed to roam in an enclosed area of 1000 square feet. The area is divided into 200 subsections of 5 square feet each.

a)Assuming that the beetles spread evenly throughout the enclosed area, how many beetles are expected to be within each subsection?

b)What is the standard deviation of x, the number of beetles in a particular subsection?

c)What is the probability of finding exactly 8 beetles in a particular subsection?

d)Would it be unusual to find more than 16 beetles in a particular subsection?

If the beetles spread evenly throughout the enclosed region, we can model the distribution of the beetles using the Poisson probabilities.

and σ=

P(8) =0.1126

About 11out of every 100 times, we expect to find exactly 8 beetles in a particular subsection.