# Probability mass function

File:Discrete probability distrib.png
The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.

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

File:Fair dice probability distribution.svg
The probability mass function of a fair die. All the numbers on the die have an equal chance of appearing on top when the die is rolled.
Probability mass function for the binomial distribution for various parameters. The lines connecting the dots are added for clarity.

Suppose that X is a discrete random variable, taking values on some countable sample space  SR. Then the probability mass function  fX(x)  for X is given by

${\displaystyle f_{X}(x)={\begin{cases}\Pr(X=x),&x\in S,\\0,&x\in \mathbb {R} \backslash S.\end{cases}}}$

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 xR\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

${\displaystyle f_{X}(x)={\begin{cases}{\frac {1}{2}},&x\in \{0,1\},\\0,&x\in \mathbb {R} \backslash \{0,1\}.\end{cases}}}$