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

2. 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

3. 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

4. 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:

5. 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.

6. 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.