Handout_7 THE POISSON DISTRIBUTION

Handout_7 THE POISSON DISTRIBUTION - May 1109 THE POISSON...

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May 11’09 THE POISSON PROBABILITY DISTRIBUTION #7 This discrete distribution, named for French mathematician S.D.Poisson (1781-1840), is used extensively as a probability model in biology and medicine. If x is a number of occurrences of some random event in an interval of time or space (or some volume of matter), the Poisson probability that x will occur is given by f(x) = e λ λ x /x! ( x = 0, 1, 2, … ) (7.1) The Greek letter λ (lambda) is called the parameter of the distribution and is the average number of occurrences of the random event in an interval (or volume). The symbol e in (7.1) is the constant 2.71828 It can be shown that f(x) ≥ 0 for every x and Σ x f(x) = 1 so the distribution (7.1) satisfies the requirements for a probability distribution. Some statistical problems require the evaluating of a cumulative Poisson probability function f(X ≤ x) which is defined as follows: f(X ≤ x) = Σ f(X) (7.1a) all X ≤ x The function (7.1a) represents the probability that a number X of occurrences of the random event of interest in an interval of time or space (or some volume of matter) does not exceed the given number x . In the expanded form, x x f(X ≤ x) = Σ f(i) = Σ e λ λ i / i! (7.1b) i=0 i=0 A sum of probabilities P(X ≥ x) + P(X ≤ x – 1 ) = 1 for both binomial and Poisson discrete probability distributions because the events X ≥ x and X ≤ x –1 are complementary to each other; also P(k ≤ X ≤ m) = P(X ≤ m) – P(X ≤ k – 1 ). An interesting feature of the Poisson distribution is the fact that both the mean µ and the variance σ 2 are equal to the parameter λ of the distribution . The Poisson process
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Handout_7 THE POISSON DISTRIBUTION - May 1109 THE POISSON...

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