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SADC Course in Statistics The Poisson distribution (Session 07)

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To put your footer here go to View > Header and Footer 2 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

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

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

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To put your footer here go to View > Header and Footer 5 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: Poisson Distribution Function

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

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

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To put your footer here go to View > Header and Footer 8 Graph of Poisson with = 15

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To put your footer here go to View > Header and Footer 9 Graph of Poisson with = 10

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To put your footer here go to View > Header and Footer 10 Graph of Poisson with = 7

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To put your footer here go to View > Header and Footer 11 Graph of Poisson with = 4

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To put your footer here go to View > Header and Footer 12 Graph of Poisson with = 1

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To put your footer here go to View > Header and Footer 13 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?

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

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To put your footer here go to View > Header and Footer 15 The expected value of the Poisson random variable (r.v.) with parameter is equal to Note that, since Poisson is a probability distribution, Expected value of a Poisson r.v.

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To put your footer here go to View > Header and Footer 16 The second moment, E(X 2 ) can be shown to be: Variance of a Poisson r.v. Hence The standard deviation of a Poisson random variable is therefore.

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To put your footer here go to View > Header and Footer 17 Cumulative probability distribution

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To put your footer here go to View > Header and Footer 18 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. Interpreting the cumulative dist n

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To put your footer here go to View > Header and Footer 19 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) = 5 15 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! Class Exercise

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To put your footer here go to View > Header and Footer 20 (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? Class Exercise – continued…

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To put your footer here go to View > Header and Footer 21 Further practical examples follow…

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