Hypergeometric, Poisson and Continuous Uniform Distributions

Lesson 29 of STA301 explores specialized probability distributions: the Hypergeometric Distribution, the Poisson Distribution, and the Continuous Uniform Distribution. While the binomial distribution requires independent trials with constant probability (sampling with replacement), the Hypergeometric Distribution models sampling without replacement from a finite population of size N containing k successes and N - k failures. When a random sample of size n is selected without replacement, the probability of obtaining x successes is given by: P(X = x) = [kCx * (N - k)C(n - x)] / [NCn]. The mean is mu = n*(k / N) and the variance is sigma^2 = n*(k / N)*(1 - k / N)*[(N - n) / (N - 1)], incorporating the Finite Population Correction factor. When population size N is large relative to sample size n (n / N <= 0.05), the hypergeometric distribution is well-approximated by the binomial distribution. The lecture investigates the Poisson Distribution, developed by Siméon Denis Poisson to model the number of rare events occurring in a continuous interval of time, space, or volume (such as typing errors per page, radioactive decay counts, or incoming call arrivals). The Poisson PMF is given by: P(X = x) = [e^(-lambda) * lambda^x] / x! for x = 0, 1, 2, ... where lambda > 0 is the average event rate and e is the mathematical constant approximately 2.71828. A defining mathematical hallmark of the Poisson distribution is that its mean and variance are identically equal: mu = sigma^2 = lambda. The Poisson distribution also serves as the limiting form of the binomial distribution when n is very large, p is very small, and the product lambda = n*p remains moderate. Finally, the Continuous Uniform (Rectangular) Distribution is examined over interval [a, b], with constant density f(x) = 1 / (b - a), mean mu = (a + b) / 2, and variance sigma^2 = (b - a)^2 / 12. In Virtual University final term examinations, Poisson rare-event word problems, verifying mean equals variance, and hypergeometric lot selections are prominent exam questions. Studying sta301 handouts and solved papers ensures thorough preparation.