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.
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. (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.
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
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.