Transcription of Chapter 4 The Poisson Distribution
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Chapter 4 The Poisson Distribution
Chapter 4 The Poisson The Fish Distribution ?The Poisson Distribution is named after Simeon-Denis Poisson (1781 1840). In addition,poissonis French for this Chapter we will study a family of probability distributions for a countably infinite samplespace, each member of which is called aPoisson Distribution . Recall that a binomial distributionis characterized by the values of two parameters:nandp. A Poisson Distribution is simpler in thatit has only one parameter, which we denote by , pronouncedtheta. (Many books and websitesuse , pronounced lambda, instead of .) The parameter must be positive: >0. Below is theformula for computing probabilities for the (X=x) =e xx!,forx= 0,1,2,3, ..( )In this equation,eis the famous number from calculus,e= limn (1 + 1/n)n= ..You might recall from the study of infinite series in calculus, that x=0bx/x! =eb,for any real numberb. Thus, x=0P(X=x) =e x=0 x/x!
The binomial distribution is appropriate for counting successes in n i.i.d. trials. For p small and n large, the binomial can be well approximated by the Poisson. Thus, it is not too surprising to learn that the Poisson is also a model for counting successes. Consider a process evolving in time in which at ‘random times’ successes occur ...
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