The Poisson distribution (Session 07)

Learning Objectives At the end of this session, you will be able to: describe the Poisson probability distribution including the underlying assumptions calculate Poisson probabilities using a calculator, or Excel software apply the Poisson model in appropriate practical situations

Examples of data on counts A common form of data occurring in practice are data in the form of counts, e.g. number of road accidents per year at different locations in a country number of children in different families number of persons visiting a given website across different days number of cars stolen in the city each month An appropriate probability distribution for this type of random variable is the Poisson distribution.

The Poisson distribution The Poisson is a discrete probability distribution named after a French mathematician Siméon-Denis Poisson, 1781-1840. A Poisson random variable is one that counts the number of events occurring within fixed space or time interval. The occurrence of individual outcomes are assumed to be independent of each other.

Poisson Distribution Function While the number of successes in the binomial distribution has n as the maximum, there is no maximum in the case of Poisson. This distribution has just one unknown parameter, usually denoted by  (lambda). The Poisson probabilities are determined by the formula:

Example: Number of cars stolen Suppose the number of cars stolen per month follows a Poisson distribution with parameter  = 3 What is the probability that in a given month Exactly 2 cars will be stolen? No cars will be stolen? 3 or more cars will be stolen?

Example: Number of cars stolen For the first two questions, you will need: = The 3rd is computed as = 1 – P(X=0) – P(X=1) – P(X=2)

Graph of Poisson with  = 15

Graph of Poisson with  = 10

Graph of Poisson with  = 7

Graph of Poisson with  = 4

Graph of Poisson with  = 1

Practical quiz What do you observe about the shapes of the Poisson distribution as the value of the Poisson parameter  increases? Approximately where does the peak of the distribution occur?

Properties of the Poisson distribution The mean of the Poisson distribution is the parameter . The standard deviation of the Poisson distribution is the square root of . This implies that the variance of a Poisson random variable = . The Poisson distribution tends to be more symmetric as its mean (or variance) increases.

Expected value of a Poisson r.v. The expected value of the Poisson random variable (r.v.) with parameter  is equal to Note that, since Poisson is a probability distribution,

Variance of a Poisson r.v. The second moment, E(X2) can be shown to be: Hence The standard deviation of a Poisson random variable is therefore  .

Cumulative probability distribution

Interpreting the cumulative distn Note that for X larger than about 12, the cumulative probability is almost equal to 1. In applications this means that, if say, the family size follows a Poisson distribution with mean 5, then it is almost certain that every family will have less than 12 members. Of course there is still the possibility of rare exceptions.

Class Exercise Answer: P(X=15) = 515 e-5/15! In example above, we assumed X=family size, has a Poisson distribution with =5. Thus P(X=x) = 5x e-5/x! , x=0, 1, 2, …etc. What is the chance that X=15? Answer: P(X=15) = 515 e-5/15! = 0.000157 This is very close to zero. So it would be reasonable to assume that a family size of 15 was highly unlikely!

Class Exercise – continued… (b) What is the chance that a randomly selected household will have family size < 2 ? To answer this, note that P(X < 2) = P(X = 0) + P(X = 1) = (c) What is the chance that family size will be 3 or more?

Further practical examples follow…