2 Learning Objectives At the end of this session, you will be able to: describe the Poisson probability distribution including the underlying assumptionscalculate Poisson probabilities using a calculator, or Excel softwareapply the Poisson model in appropriate practical situations
3 Examples of data on counts A common form of data occurring in practiceare data in the form of counts, e.g.number of road accidents per year at different locations in a countrynumber of children in different familiesnumber of persons visiting a given website across different daysnumber of cars stolen in the city each monthAn appropriate probability distribution for thistype of random variable is the Poissondistribution.
4 The Poisson distribution The Poisson is a discrete probability distribution named after a French mathematician Siméon-Denis Poisson,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.
5 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:
6 Example: Number of cars stolen Suppose the number of cars stolen per month follows a Poisson distribution with parameter = 3What is the probability that in a given monthExactly 2 cars will be stolen?No cars will be stolen?3 or more cars will be stolen?
7 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)
13 Practical quizWhat 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?
14 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.
15 Expected value of a Poisson r.v. The expected value of the Poisson random variable (r.v.) with parameter is equal toNote that, since Poisson is a probability distribution,
16 Variance of a Poisson r.v. The second moment, E(X2) can be shown to be:HenceThe standard deviation of a Poisson random variable is therefore .
18 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.
19 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!=This is very close to zero. So it would be reasonable to assume that a family size of 15 was highly unlikely!
20 Class Exercise – continued… (b) What is the chance that a randomly selected household will have family size < 2 ?To answer this, note thatP(X < 2) = P(X = 0) + P(X = 1)=(c) What is the chance that family size will be 3 or more?