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In probability theory, a **probability mass function** (abbreviated **pmf**) gives the probability that a discrete random variable is exactly equal to some value. A probability mass function differs from a probability density function in that the values of the latter, defined only for continuous random variables, are not probabilities; rather, its integral over a set of possible values of the random variable is a probability.

## Mathematical descriptionEdit

Suppose that *X* is a discrete random variable, taking values on some countable sample space *S* ⊆ **R**. Then the probability mass function *f*_{X}(*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 *f*_{X}(*x*) for all real numbers, including all values in **R** that *X* could never take; indeed, it assigns such values a probability of zero. (Alternatively, think of Pr(*X* = *x*) as 0 when *x* ∈ **R**\*S*.)

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* ∈ **R**\*S*) the derivative is zero, just as the probability mass function is zero at all such points.

## ExamplesEdit

A simple example of a probability mass function is the following. 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 just 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} $

Probability mass functions may also be defined for any discrete random variable, including constant, binomial (including Bernoulli), negative binomial, Poisson, geometric and hypergeometric random variables.nl:kansfunctie

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