The Poisson Distribution

Contents

Introduction  Poisson Experiment  Poisson Distribution  Distributional Properties  Poisson Probabilities  Poisson Moments  Poisson Quantiles  Poisson Approximation to the Binomial  Examples 

Learning Objectives

The Poisson module targets the following cognitive tasks:

Task        Skills Concepts
Pois-1: Identify a Poisson event Understand Poisson events
Pois-2: Identify a Poisson experiment Understand the conditions of a Poisson experiment
Pois-3: Distinguish between a binomial and Poisson experiment Understand the distinction between a binomial and Poisson experiment
Pois-4: Identify a Poisson random variable Understand Poisson random variables
Pois-5: Compute Poisson probabilities Develop the Poisson probability density function (pdf)
Pois-6: Compute the Poisson mode Understand the properties of the Poisson mode
Pois-7: Approximate Poisson probabilities using the normal distribution Understand how and when normal probabilities approximate Poisson probabilities
Pois-8: Compute the theoretical mean of a Poisson random variable Derive and interpret the theoretical mean of a Poisson random variable
Pois-9: Compute the theoretical variance of a Poisson random variable Derive the theoretical variance of a Poisson random variable
Pois-10: Understand the dependence of the Poisson variance on the mean
Pois-11: Compute Poisson cumulative probabilities Develop the Poisson cumulative distribution function (cdf)
Pois-12: Compute Poisson quantiles Understand Poisson quantiles
Pois-13: Approximate binomial probabilities using the Poisson Understand how and when Poisson probabilities approximate binomial probabilities

^ Introduction

Data sometimes arise as the number of occurrences () of an event per unit time or space, e.g., the number of yeast cells per cm2 on a microscope slide. Under certain conditions (see below), the random variable is said to follow a Poisson distribution, which, as a count, is type of discrete distribution. Occurrences are sometimes called arrivals when they take place in a fixed time interval.

The Poisson distribution was discovered in 1838 by Simeon-Denis Poisson as an approximation to the binomial distribution, when the probability of success is small and the number of trials is large. The Poisson distribution is called the law of small numbers because Poisson events occur rarely even though there are many opportunities for these evens to occur.

^ Poisson Experiment

The number of occurrences of an event per unit time or space will have a Poisson distribution if:

^ Poisson Distribution

The Poisson probability density function is given by:


where is the mean of the Poisson random variable, i.e., the average number of occurrences of the event per unit of time or space. As such, is the rate of occurrence per unit time or space. For example, if one decay event of a radioactive substance occurs per second, then .

The following Poisson Applet can be used to compute Poisson probabilites and quantiles. (Note: in the applet.)

^ Distributional Properties

Notice that the modes of the Poisson probability distribution in the applet (when ) are 3 and 2. This is always the case: if is a positive integer, the modes are: . If is not an integer, then the mode is: , which is the largest integer . Double click on m in the center of the slider and change the increment to 0.5. Animate the probability density function to see that the defintions of the mode hold.

The Poisson distribution with sufficiently large can be approximated by a Normal distribution. The approximation is good if and a continuity correction is used. For example, the Poisson probability:

,
where the later is computed from the normal distribution with a non-negative integer. The Poisson applet works for . Thus, for , the normal approximation must be used. Notice that as m in the applet increases, the distribution becomes more symmetric.

^ Poisson Probabilities

Poisson probabilities can be computed directly from the probability density function or from Poisson probability tables for certain values of . We start by verifying the rather surprising fact pointed out in the definition of the mode: , when is a positive integer.


When :

Cumulative probabilites can be computed by:

for non-negative .

When :

.

^ Poisson Moments

The mean of a Poisson random variable is , i.e., . The mean is a rate, e.g., a temporal rate for time events. For example, if 0.5 phone calls per hour are received on a home phone during the day, then the mean number of phone calls between 9 A.M. and 5 P.M. is .

The Poisson has the interesting property that the variance is also , i.e., . Thus, unlike the normal distribution, the variance of a Poisson random variable depends on the mean. Certain Poisson-like random variables are over-dispersed () or under-dispersed (). For example, the negative binomial can be viewed as an over-dispersed Poisson and, like the Poisson, is often used to model species abundances in ecology, i.e, certain species abundances are Poisson distributed and others are distributed as a negative binomial.

The mean, and hence the variance, can be estimated by the sample mean, which is the maximum likelihood estimator, i.e., if are realizations of a Poisson experiment, the estimated mean is:

.

For example, suppose a supervisor wants to know the average number of typing mistakes his/her secretary makes per page. Ten pages are rondomly selected and the the following values were obtained:

2 1 3 1 3 3 3 2 3 1
Then:
.
The value of actually used to simulate the data was 2 and thus the estimate is reasonably close.

^ Poisson Quantiles

^ Poisson Approximation to the Binomial

Consider a binomial distribution consisting of trials with probability of success , i.e,. for . If is sufficiently large, then the binomial probability:

,

i.e., the binomial probability is approximately equal to the corresponding Poisson probability. Note that , the mean of the binomial distribution.

Thus, binomial probabilities, which are hard to compute for large , can be approximated by corresponding Poisson probabilies. For example, suppose 10,000 soldiers are screened for a rare blood disease (). We want the probability at least 10 soldiers test positive for the disease, i.e, we want , where . This is difficult to compute using the binomial distribution, but much easier for the Poisson with .

^ Examples

1. Horse Kick Deaths

Self-test