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Week of September 20, 2005 Multinomial Extension of Binomial when more than two outcomes are possible. Setting: Consider a sequence of trials for which: - # of trials (n) is fixed - k possible outcomes ( E1 , E2 , are possible per trial , Ek ) - the probabilities of outcomes E1 , E2 , , Ek stay constant across trials (on a given trial, P ( Ei ) pi ) - the trials are independent Multinomial probability function gives P ( E1 occurs x1 times, E2 occurs x2 times, Ek occurs xk times ) where x1 x2 xk n Multinomial Probability Function: f ( x1 , x2 , , xk ; p1, p2 , , pk , n) n x1 x2 x , x , , x p1 p2 k 1 2 where x1 x2 xk n p1 p2 pk 1 and n n! x , x , , x x !x ! x ! k 1 2 1 2 k pkxk Notes: (1) Binomial is special case of k 2 (2) pk 1 p1 pk 1 i.e. the probabilities are completely specified by pi , i 1, , k 1 (in binomial case, we only had to specify p ) Note: Binomial and Multinomial Sampling is WITH Replacement - when a “trial” consists of sampling from a population, the binomial is applicable to sampling WITH replacement (this gives independent outcomes -- probabilities don’t change) - binomial is a good approximation for sampling without replacement when # of trials (observations) is small with respect to population size from which the sample is taken Hypergeometric Setting: 1. Random sample of size n is selected without replacement from N items 2. Two types of items among the N -- k of one type (“success”) -- N k are of the other type (“failure”) 3. We want to know the probability of getting x successes out of the n objects selected. Hypergeometric Probability Function: k N k x n x , h( x; N , n, k ) N n x 0,1, ,n Mean and Variance of Hypergeometric: nk N N n k k n (1 ) N 1 N N 2 Relationship between Hypergeometric and Binomial - if N is large with respect to n, then these formulas become very similar to those for the binomial We will not cover: - Multivariate Hypergeometric (page 130) - Negative Binomial and Geometric (sec 5.5) Poisson Process Assume that events occur at random throughout an interval such that: 1. The # of outcomes in any one interval is independent of the # occurring in any disjoint interval 2. The probability of a single occurrence in a short interval is proportional to the length of the interval 3. The probability that > 1 outcome will occur in a short interval is negligible Poisson Distribution X = # of occurrences in an interval of length t during a Poisson experiment. - the probability function for X is e t ( t ) x p( x; t ) , x 0,1,2, x! where (>0) is the mean number of outcomes in a unit interval Mean and Variance of Poisson: t 2 t Poisson Tables (Table A.2 – pp. 667-) Tabled values are Poisson probability sums r P ( X r ) p ( x; t ) x 0 Notes about Table A.2: (1) Same format as Binomial table (A.1) (2) Tabled value is P ( X r ) Note Also: Poisson probabilities can be thought of as limits of Binomial probabilities when n , p 0 in such a way that np stays constant - i.e. binomial probabilities can be approximated using Poisson when n is large and p is small