Discretevalued

Discrete-valued refers to a quantity, typically a random variable, that takes values in a countable set, usually integers, rather than forming a continuous range. In probability theory, a discrete-valued random variable X is described by a probability mass function p(k) = P(X = k), defined on the set of values that X can assume, called the support. The cumulative distribution function F(x) = P(X ≤ x) for a discrete-valued variable is a step function with jumps at the values in the support.

The moments of a discrete-valued variable are computed from the probability mass function. The expected value is E[X] = sum over all k of k p(k), and the variance is Var(X) = E[X^2] − (E[X])^2, where E[X^2] = sum k^2 p(k). Convergence of these sums is required for finite moments. Commonly encountered discrete-valued distributions include Bernoulli, Binomial, Poisson, Geometric, and Negative Binomial. The probability generating function G(t) = E[t^X] = sum p(k) t^k is a useful tool for analyzing such variables, offering a compact way to derive moments and distributional properties.

Discrete-valued models are appropriate for count data and measurements inherently restricted to whole units, such as the number of emails per day, defect counts, or arrivals in a queue. They contrast with continuous-valued variables, which can take any value in an interval and are described by probability density functions. Mixed distributions can occur when a variable has both discrete atoms and a continuous component.