Math 4030 – 4a Discrete Distributions Binomial Hypergeometric Poisson Geometric

Hypergeometric Distribution (Sec. 4.3) There are N units, of which a units are defective. Randomly sample n units without replacement, and let X be the number of defective units in the sample. Then Sampling without replacement, and N is not large enough. 4/24/2017

Probability P(X=x) = h(x; n, a, N)? Mean: Variance: 4/24/2017

Binomial vs. Hypergeometric N - a n - x x a In a pool of N objects, a are marked with an “S”. Randomly select n. X is the number of S’s in the sample of n. X ~ H(n, a, N) All possible values for X: In an infinitely large pool, p100% are marked with an “S”. Randomly select n. X is the number of S’s in the sample of n. X ~ Bi(n, p) All possible values for X: X = 0, 1, 2, …, n 0  x  n 0  x  a 0  n – x  N - a 4/24/2017

Binomial vs. Hypergeometric X ~ Bi(n, p) X ~ H(n, a, N) 4/24/2017

Poisson Distribution (Sec. 4.6) 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 4/24/2017

Poisson Processes (Sec. 4.7) 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 4/24/2017

Geometric Distribution (Sec. 4.8) 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? Or Geometric? 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.

Chebyshev’s Theorem (Sec.4.5) If X is a random variable with mean  and standard deviation , then for any number k > 1, Probability Define the boundaries based on SD Central location the distribution 4/24/2017

Application? The number of customers who visit a car dealer’s showroom is a random variable with mean 18 and standard deviation 2.5. With what probability can we assert that there will be more than 8 but fewer than 28 customers? 4/24/2017