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# Probability mass function

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 description

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

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. (Alternatively, think of  Pr(X = x)  as 0 when  xR\\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 xR\\S) the derivative is zero, just as the probability mass function is zero at all such points.

## Examples

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.