Probability mass function

In probability theory, a probability mass function (abbreviated pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. A probability mass function differs from a probability density function (abbreviated pdf) in that the values of a pdf, defined only for continuous random variables, are not probabilities as such. Instead, the integral of a pdf over a range of possible values (a, b ] gives the probability of the random variable falling within that range.

Mathematical description
Suppose that X is a discrete random variable, taking values on some countable sample space S &sube; R. Then the probability mass function fX(x)  for X is given by
 * $$f_X(x) = \begin{cases} \Pr(X = x), &x\in S,\\0, &x\in \mathbb{R}\backslash S.\end{cases}$$

Note that this explicitly defines fX(x)  for all real numbers, including all values in R that X could never take; indeed, it assigns such values a probability of zero.

The discontinuity of probability mass functions reflects the fact that the cumulative distribution function of a discrete random variable is also discontinuous. Where it is differentiable (i.e. where x &isin; R\S) the derivative is zero, just as the probability mass function is zero at all such points.

Example
Suppose that X is the outcome of a single coin toss, assigning 0 to tails and 1 to heads. The probability that X = x is 0.5 on the state space {0, 1} (this is a Bernoulli random variable), and hence the probability mass function is
 * $$f_X(x) = \begin{cases}\frac{1}{2}, &x \in \{0, 1\},\\0, &x \in \mathbb{R}\backslash\{0, 1\}.\end{cases}$$

Wahrscheinlichkeitsfunktion Variable aléatoire kansfunctie Função massa de probabilidade Функция вероятности Funzione di probabilità 概率质量函数