Math 4030 – 4a More Discrete Distributions

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Presentation transcript:

Math 4030 – 4a More Discrete Distributions Poisson Geometric Negative Binomial

Poisson Distribution Random variable X follows Poisson distribution, or X ~ Poisson(), if its probability distribution has the following formula where  > 0 is the parameter (both mean and variance). The cumulative distribution 5/10/2018

Poisson Processes Consider independent and random “customer” arrivals over a given time interval. Let X be the number of arrivals. Then X has the Poisson distribution with parameter  as the mean/average number of arrivals on the interval. phone calls at customer service; students’ visit during office hour; machine breakdown; forest fire; earthquake 5/10/2018

Geometric Distribution A quality inspector inspects the electrical switches right off the manufacturing belt. He is interested in the question: How may items are to be inspected until the first failure occurs? P(X=x) = g(x;p) = p(1-p)x-1, for x = 1,2,…. Cumulative probability: The mean and the variance:

Negative Binomial Distribution NB(r,p): A quality inspector inspects the electrical switches right off the manufacturing belt. He is interested in the question: How may items are to be inspected until the r failures are found? When r = 1, we have geometric distribution.

Binomial? Hypergeometric? Poisson? Geometric? Or Negative Binomial? Binomial – Sample with replacement: n trials, each with two outcomes (S or F), identical probability p, independent. X is the number of “S” in n trials. Hypergeometric – Sample without replacement: N objects, of them a have marked with “S”. Take a sample of size n (without replacement). X is the number of “S” in the sample. Poisson – Number of arrivals: In a given time period, there are  independent arrivals on average. X is the actual number of arrivals in any given time period. Geometric – When to get the first “S”: Repeat the independent Bernoulli trials until the first “S” occurs. X is the number of trials repeated. Negative Binomial – When to get the r-th “S”: Repeat the independent Bernoulli trials until the exactly r “S” occurs. X is the number of trials repeated.