# 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* ⊆ **R**. Then the probability mass function *f*_{X}(*x*) for *X* is given by

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.

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.

## 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

de:Wahrscheinlichkeitsfunktion nl:kansfunctie it:Funzione di probabilità