Discrete Probability Distributions
Several standard discrete distributions arise repeatedly across computing, reliability, and data analysis. Knowing their forms, parameters, and typical applications saves a great deal of effort in modelling.
Bernoulli Distribution
A Bernoulli trial has two outcomes, success with probability p and failure with probability 1 − p. The random variable X takes value 1 for success, 0 for failure. Mean is p and variance is p(1 − p). Bernoulli trials are the atomic building blocks.
Binomial Distribution
The binomial distribution counts the number of successes in n independent Bernoulli trials with success probability p. Its PMF is P(X = k) = C(n, k) pk (1 − p)n − k. Mean is np and variance is np(1 − p). It models coin tossing, quality control, and A/B testing.
Geometric Distribution
The geometric distribution counts the number of Bernoulli trials until the first success. P(X = k) = (1 − p)k − 1 p for k = 1, 2, .... Mean is 1/p and variance is (1 − p)/p2. It is memoryless: P(X > m + n | X > m) = P(X > n), a property shared only with the exponential distribution.
Negative Binomial Distribution
The negative binomial generalizes the geometric: it counts the number of trials needed to get r successes. It is often parameterized alternatively as the number of failures before the rth success. Applications include overdispersed count data where the Poisson is too restrictive.
Hypergeometric Distribution
The hypergeometric distribution counts the number of successes when drawing without replacement from a finite population of K successes and N − K failures. Unlike the binomial, trials are not independent. Used in sampling inspection and card probabilities.
Poisson Distribution
The Poisson distribution counts the number of events in a fixed interval when events occur independently at constant average rate λ. P(X = k) = e−λ λk / k!. Mean and variance are both λ. Poisson approximates binomial when n is large and p is small (np ≈ λ). It models arrivals, telephone calls, server requests, and rare events.
Discrete Uniform Distribution
The discrete uniform distribution assigns equal probability 1/n to each of n values. It underlies dice rolls, random permutations, and the fairness analysis of algorithms.
Multinomial Distribution
The multinomial extends the binomial to more than two categories. For n independent trials each falling into category i with probability pi, the joint PMF describes how the n outcomes are distributed. Used in text classification and categorical data analysis.
Relations Among Distributions
Sums of independent binomials with the same p are binomial. Sums of independent Poissons with rates λi are Poisson with rate Σ λi. A binomial with small p and large n approaches Poisson; the Poisson itself approaches a normal for large λ.
Summary
Discrete distributions like Bernoulli, binomial, geometric, hypergeometric, and Poisson describe many real-world counting processes. Recognizing which distribution fits a problem, and recalling its moments and approximations, is a routine skill in applied statistics.