Discrete Random Variables: Summary of the Discrete Probability Functions 1.5 2008/05/29 13:37:56 GMT-5 2008/07/31 10:13:56.470 GMT-5 Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Connexions cnx@cnx.org binomial discrete elementary formula function geometric hypergeometrical Poisson probability random statistics variable This module provides a review of the binomial, geometric, hypergeometric, and Poisson probability distribution functions and their properties. Binomial X~B(n,p)X = the number of successes n = the number of independent trials X takes on the values x= 0,1, 2, 3, ...,n p = the probability of a success q = the probability of a failure p+q=1 q=1-p The mean is μ=np. The variance is σ2=npq. Geometric X~G (p) X = the number of trials until the first success (count the failures and the first success) X takes on the values x= 1, 2, 3, ... p = the probability of a success q = the probability of a failure p+q=1 q=1-p The mean is μ= 1p Τhe variance is σ2 = 1p (( 1p) -1) Hypergeometric X~H (r, b, n)X = the number of items from the group of interest that are in the chosen sample. X may take on the values x= 0, 1, ..., up to the size of the group of interest. (The minimum value for X may be larger than 0 in some instances.) r = the size of the group of interest (first group) b= the size of the second group n= the size of the chosen sample. n r + b The mean is: μ= n r r + b The variance is: σ2 r b n ( r + b + n ) ( r + b ) 2 ( r + b - 1 ) Poisson X ~ P(μ)X = the number of occurrences in the interval of interestX takes on the values x = 0, 1, 2, 3, ...The mean μ is typically given. (λ is often used as the mean instead of μ.) When the Poisson is used to approximate the binomial, we use the binomial mean μ = np. n is the binomial number of trials. p = the probability of a success for each trial. This formula is valid when n is "large" and p "small". n "large" is about 100 and p "small" may be something less than 0.1. If n is large enough and p is small enough then the Poisson approximates the binomial very well.