massifunktsioon

Massifunktsioon, known in English as the probability mass function (PMF), describes the distribution of a discrete random variable. If X takes values in a countable set S, the massifunktsioon is defined by p_X(x) = P(X = x) for x in S. The values satisfy p_X(x) ≥ 0 for all x and sum_{x in S} p_X(x) = 1. For values outside S, p_X(x) = 0.

The massifunktsioon uniquely determines the distribution of X. The cumulative distribution function F_X(x) = P(X ≤ x) can be obtained by summing p_X(t) over t ≤ x. For finite or countably infinite S, the sum converges to 1.

Moments of X can be computed from the massifunktsioon: the mean is E[X] = sum_{x in S} x p_X(x), and the variance is Var(X) = E[X^2] − (E[X])^2, provided the sums converge.

Examples: Bernoulli(p) has p_X(1) = p, p_X(0) = 1−p. Binomial(n,p) has p_X(k) = C(n,k) p^k (1−p)^{n−k} for k = 0,1,...,n. Poisson(λ) has p_X(k) = e^{−λ} λ^k / k! for k ≥ 0.

Joint massifunktsioon p_{X,Y}(x,y) = P(X=x, Y=y) describes the distribution of multiple discrete variables; the sum over all x,y equals 1. The massifunktsioon is central to likelihood-based inference for discrete data and to computing moments and probabilities for discrete models.