Probability mass function

File:Discrete probability distrib.png

The graph of a 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; rather, its integral over a set of possible values of the random variable is a probability.

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

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

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.

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 USMA Math Department is evil. 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

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