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Discrete Random Variables: Poisson (optional)

Module by: Susan Dean, Dr. Barbara Illowsky

Summary: This module describes the characteristics of a Poisson experiment and the Poisson probability distribution. This module is included in the Elementary Statistics textbook/collection as an optional lesson.

Characteristics of a Poisson experiment are:

  1. You are interested in the number of times something happens in a certain interval. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are 5 words spelled incorrectly in 100 pages. The interval is the 100 pages.
  2. The Poisson may be derived from the binomial if the probability of success is "small" (such as 0.01) and the number of trials is "large" (such as 1000). You will verify the relationship in the homework exercises. nn is the number of trials and pp is the probability of a "success."
The outcomes of a Poisson experiment fit a Poisson probability distribution. The mean and variance are given in the summary of this chapter.

Example 1

The average number of loaves of bread put on a shelf in a bakery in a half-hour period is 12. What is the probability that the number of loaves put on the shelf in 5 minutes is 3? Of interest is the number of loaves of bread put on the shelf in 5 minutes. The time interval of interest is 5 minutes.

Let XX = the number of loaves of bread put on the shelf in 5 minutes. If the average number of loaves put on the shelf in 30 minutes (half-hour) is 12, then the average number of loaves put on the shelf in 5 minutes is

( 530 )⋅12=2 ( 530 )⋅12 2 loaves of bread

The probability question asks you to find P(X = 3)P(X = 3).

Example 2

A certain bank expects to receive 6 bad checks per day. What is the probability of the bank getting fewer than 5 bad checks on any given day? Of interest is the number of checks the bank receives in 1 day, so the time interval of interest is 1 day. Let XX = the number of bad checks the bank receives in one day. If the bank expects to receive 6 bad checks per day then the average is 6 checks per day. The probability question asks for P(X<5) P(X 5) .

Example 3

Your math instructor expects you to complete 2 pages of written math homework every day. What is the probability that you complete more than 2 pages a day?

This is a Poisson problem because your instructor is interested in knowing the number of pages of written math homework you complete in a day.

Problem 1

What is the interval of interest?

Solution 1

The 2 pages

Problem 2

What is the average number of pages you should do in one day?

Solution 2

2

Problem 3

Let XX = ____________. What values does XX take on?

Solution 3

Let XX = the number of pages of written math homework you do per day.

Problem 4

The probability question is P(______)P(______).

Solution 4

P(X > 2)P(X > 2)

Notation for the Poisson: P = Poisson Probability Distribution Function

X X ~ P(μ) P(μ)

Read this as "XX is a random variable with a Poisson distribution." The parameter is μμ (or λλ). μμ (or λλ) = the mean for the interval of interest.

Example 4

Leah's answering machine receives about 6 telephone calls between 8 a.m. and 10 a.m. What is the probability that Leah receives more than 1 call in the next 15 minutes?

Let XX = the number of calls Leah receives in 15 minutes. (The interval of interest is 15 minutes or 1414 hour.)

XX takes on the values 0, 1, 2, 3, ...

If Leah receives, on the average, 6 telephone calls in 2 hours, and there are eight 15 minutes intervals in 2 hours, then Leah receives

18 ⋅6 =0.75 18 ⋅6 0.75

calls in 15 minutes, on the average. So, μμ = 0.75 for this problem.

XX ~ P(0.75)P(0.75)

Find P(X>1) P(X 1). P(X>1)=0.1734 P(X 1) 0.1734 (calculator or computer)

TI-83+ and TI-84: For a general discussion, see this example (Binomial). The syntax is similar. The Poisson parameter list is (μμ for the interval of interest, number). For this problem:

Press 1- and then press 2nd DISTR. Arrow down to C:poissoncdf. Press ENTER. Enter .75,1). The result is P(X>1)=0.1734 P(X 1) 0.1734. NOTE: The TI calculators use λλ (lambda) for the mean.

The probability that Leah receives more than 1 telephone call in the next fifteen minutes is about 0.1734.

The graph of XX ~ P(0.75)P(0.75) is:

The poisson probability distribution function graph has 5 bars decreasing from left to right with an x-axis of 0-∞ and a y-axis of 0-0.5 in increments of 0.1. The x-axis is equal to the number of calls Leah receives within 15 minutes.

The y-axis contains the probability of XX where XX = the number of calls in 15 minutes.

Glossary

Poisson Distribution:
A discrete random variable (RV) is the number of times a certain event will occur in a specific period of time, or in specific area, or any other units of measurement. The characteristics of the variable are: the probability that an event occurs in a given unit is the same for all units and doesn’t depend on the number of event that occurs in the other units. The distribution is completely defined by the mean number μ of event in the unit interval of measurement. The notation is: X~P(μ)X~P(μ) size 12{X "~" P \( μ \) } {}; the domain is whole numbers, {0,1,2,...}{0,1,2,...} size 12{ lbrace 0,1,2, "." "." "." rbrace } {}; the mean is μ=npμ=np size 12{μ= ital "np"} {}, and the variance is σ 2 = μ 2 σ 2 = μ 2 , the probability to have exactly xx successes in rr trials is P(X=x)=e−μμxx!P(X=x)=e−μμxx! size 12{P \( X=x \) =e rSup { size 8{ - μ} } { {μ rSup { size 8{x} } } over {x!} } } {}. The Poisson distribution often used to approximate the binomial distribution when n is “large” and p is “small” (a general rule is that n should be equal to or greater than 20 and pp equal to or less than .05).

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