THE POISSON DISTRIBUTION The Poisson random variable was first introduced by the French mathematician Simeon-Denis Poisson (1781-1840). He discovered it as a limit to the binomial distribution as the number of trials n approaches infinity. The Poisson distribution is used to determine the probability of obtaining a certain number of successes within a certain interval (of time or space).

Introduction to the Poisson Distribution Poisson distribution is a distribution of a discrete random variable X—if events happen at a constant rate over time, the Poisson distribution gives the probability of X number of events occurring within a certain interval (of time or space). Possible examples are when describing the number of flaws in a given length of material; accidents on a particular stretch of road in a week, telephone calls made to a switchboard in one day; the number of misprints on a typical page of a book; the number of fish caught in a large lake per day.

Poisson Distribution, example The probability distribution function for the discrete Poisson random variable is where m is the parameter of the distribution.

Formula Booklet the parameter is called m. Poisson distribution

Poisson Mean and Variance For a Poisson random variable, the variance and mean are the same! Mean Variance and Standard Deviation where m = expected number of hits in a given time period, it is called the parameter.

Example The number of accidents in a randomly chosen day at a road crossing can be modelled by a Poisson distribution with parameter 0.5. Find the probability that, on a randomly chosen day, there are: a) exactly two accidents b) at least two accidents

Using Poisson pdf on GDC, part a: P(X=2)=0.0758 (3 sf)

Part b) Using Poisson cdf on GDC:

Example For example, if new cases of West Nile Virus in New England are occurring at a rate of about 2 per month, then these are the probabilities that: 0,1, 2, 3, 4, 5, 6, to 1000 to 1 million to… cases will occur in New England in the next month:

Poisson Probability table x P(X=x) =.135 1 =.27 2 3 =.18 4 =.09 5   …

Using GDC Poisson pdf with the list:

Example: Poisson distribution Suppose that a rare disease has an incidence of 1 in 1000 person-years. Assuming that members of the population are affected independently, find the probability of k cases in a population of 10,000 (followed over 1 year) for k=0,1,2.   The expected value (mean) = = .001*10,000 = 10 10 new cases expected in this population per year.

Finding the parameter If find the value of the parameter correct to 4 decimal places, given that Using Numerical solve on GDC:

Mean and variance of Poisson distribution If and X is defined on an interval of length l, to find the probability in an interval of length kl we use another Poisson distribution with mean

Example In a book of 520 pages there are 135 misprints. What is the mean number of misprints per page? Find the probability that pages 444 and 445 do not contain any misprints?

Let X be the number of misprints found on 2 pages. We need to find P(X=0), using GDC:

more on Poisson… “Poisson Process” (rates) Note that the Poisson parameter  can be given as the mean number of events that occur in a defined time period OR, equivalently,  can be given as a rate, such as =2/month (2 events per 1 month) that must be multiplied by t=time (called a “Poisson Process”)  X ~ Poisson () E(X) = t Var(X) = t

Example For example, if new cases of West Nile in New England are occurring at a rate of about 2 per month, then what’s the probability that exactly 4 cases will occur in the next 3 months? X ~ Poisson (=2/month) Exactly 5 cases?

On Sunday mornings, cars arrive at a petrol station at an average rate of 30 per hour. Assuming that the number of cars arriving at the petrol station follows a Poisson distribution, find the probability that: in a half-hour period 12 cars arrive no cars arrive during a particular 5-minute interval more than 4 cars arrive in a 15 minute interval We need to calculate a new parameter for each time interval as follows: 0.5 times 30 = 15 5 times 30/60=2.5 15 times 30/60=7.5 And find each probability using GDC:

Practice problems If calls to your cell phone are a Poisson process with a constant rate =2 calls per hour, what’s the probability that, if you forget to turn your phone off in a 1.5 hour movie, your phone rings during that time?

Answer If calls to your cell phone are a Poisson process with a constant rate =2 calls per hour, what’s the probability that, if you forget to turn your phone off in a 1.5 hour movie, your phone rings during that time? X ~ Poisson (=2 calls/hour) P(X≥1)=1 – P(X=0)

The random variable X follows the Poisson distribution with mean m and it satisfies the following equation: Find the value of m correct to 4 decimal places Calculate

a) Using the formula we can write the following equation: Now use numerical solve on GDC:

b) Using GDC with storing the value of m: