The Poisson distribution SADC Course in Statistics The

The Poisson distribution SADC Course in Statistics

The Poisson Distribution • Apply the Poisson Distribution when: – You wish to count the number of times an event occurs in a given area of opportunity – The probability that an event occurs in one area of opportunity is the same for all areas of opportunity – The number of events that occur in one area of opportunity is independent of the number of events that occur in the other areas of opportunity – The probability that two or more events occur in an area of opportunity approaches zero as the area of opportunity becomes smaller To put your footer here go to View > Header and Footer

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. To put your footer here go to View > Header and Footer 3

The Poisson Distribution The Poisson distribution is defined by: Where f(x) is the probability of x occurrences in an interval m is the expected value or mean value of occurrences within an interval e is the natural logarithm. e = 2. 71828 To put your footer here go to View > Header and Footer

Properties of the Poisson Distribution 1. The probability of occurrences is the same for any two intervals of equal length. 2. The occurrence or nonoccurrence of an event in one interval is independent of an occurrence on nonoccurrence of an event in any other interval To put your footer here go to View > Header and Footer

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. To put your footer here go to View > Header and Footer 6

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. To put your footer here go to View > Header and Footer 7

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: To put your footer here go to View > Header and Footer 8

Example: Mercy Hospital • Poisson Probability Function Patients arrive at the MERCY emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? To put your footer here go to View > Header and Footer

Example: Mercy Hospital n Poisson Probability Function = 6/hour = 3/half-hour, x = 4 To put your footer here go to View > Header and Footer MERCY

The Poisson Distribution, Working with the Equation • Example: During the 12 p. m. – 1 p. m. noon hour, arrivals at a curbside banking machine have been found to be Poisson distributed with a mean of 1. 3 persons per minute. If x = number of arrivals per minute, determine P(X < 2): – P(x < 2) = 0. 2725 + 0. 3543 = To put your footer here go to View > Header and Footer 0. 6268

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? To put your footer here go to View > Header and Footer 12

Example: Number of cars stolen For the first two questions, you will need: = = The 3 rd is computed as = 1 – P(X=0) – P(X=1) – P(X=2) = To put your footer here go to View > Header and Footer 13

Class Exercise In example above, we assumed X=family size, has a Poisson distribution with =5. Thus P(X=x) = 5 x e-5/x! , x=0, 1, 2, …etc. (a)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! To put your footer here go to View > Header and Footer 14

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? To put your footer here go to View > Header and Footer 15