Discrete Distributions

The functions described here are among the most commonly used discrete statistical distributions. You can compute their densities, means, variances, and other related properties. The distributions themselves are represented in the symbolic form name[param₁, param₂, ...]. Functions such as Mean, which give properties of statistical distributions, take the symbolic representation of the distribution as an argument. "Continuous Distributions" describes many continuous statistical distributions.

Most of the common discrete statistical distributions can be understood by considering a sequence of trials, each with two possible outcomes, for example, success and failure.

The Bernoulli distribution BernoulliDistribution[p] is the probability distribution for a single trial in which success, corresponding to value 1, occurs with probability p, and failure, corresponding to value 0, occurs with probability 1-p.

The binomial distribution BinomialDistribution[n, p] is the distribution of the number of successes that occur in n independent trials, where the probability of success in each trial is p.

The negative binomial distribution NegativeBinomialDistribution[n, p] for positive integer n is the distribution of the number of failures that occur in a sequence of trials before n successes have occurred, where the probability of success in each trial is p. The distribution is defined for any positive n, though the interpretation of n as the number of successes and p as the success probability no longer holds if n is not an integer.

The beta binomial distribution BetaBinomialDistribution[, , n] is a mixture of binomial and beta distributions. A BetaBinomialDistribution[, , n] random variable follows a BinomialDistribution[n, p] distribution, where the success probability p is itself a random variable following the beta distribution BetaDistribution[, ]. The beta negative binomial distribution BetaNegativeBinomialDistribution[, , n] is a similar mixture of the beta and negative binomial distributions.

The geometric distribution GeometricDistribution[p] is the distribution of the total number of trials before the first success occurs, where the probability of success in each trial is p.

The hypergeometric distribution HypergeometricDistribution[n, n_succ, n_tot] is used in place of the binomial distribution for experiments in which the n trials correspond to sampling without replacement from a population of size n_tot with n_succ potential successes.

The discrete uniform distribution DiscreteUniformDistribution[{i_min, i_max}] represents an experiment with multiple equally probable outcomes represented by integers i_min through i_max.

The Poisson distribution PoissonDistribution[] describes the number of events that occur in a given time period where is the average number of events per period.

The terms in the series expansion of log (1-) about =0 are proportional to the probabilities of a discrete random variable following the logarithmic series distribution LogSeriesDistribution[]. The distribution of the number of items of a product purchased by a buyer in a specified interval is sometimes modeled by this distribution.

The Zipf distribution ZipfDistribution[], sometimes referred to as the zeta distribution, was first used in linguistics and its use has been extended to model rare events.

Distributions are represented in symbolic form. PDF[dist, x] evaluates the mass function at x if x is a numerical value, and otherwise leaves the function in symbolic form whenever possible. Similarly, CDF[dist, x] gives the cumulative distribution and Mean[dist] gives the mean of the specified distribution. For a more complete description of the various functions of a statistical distribution, see the description of their continuous analogues in "Continuous Distributions".