Probability mass function

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 and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

A probability mass function differs from a probability density function (pdf) in that the latter is associated with continuous rather than discrete random variables; the values of the latter are not probabilities as such: a pdf must be integrated over an interval to yield a probability.

Formal definition

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 stops rolling.

Suppose that X: S → A (A $\subseteq$ R) is a discrete random variable defined on a sample space S. Then the probability mass function f_X: A → [0, 1] for X is defined as

$f_{X}(x)=\Pr(X=x)=\Pr(\{s\in S:X(s)=x\}).$

Thinking of probability as mass helps avoiding mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes x:

$\sum _{x\in A}f_{X}(x)=1$

When there is a natural order among the hypotheses x, it may be convenient to assign numerical values to them (or n-tuples in case of a discrete multivariate random variable) and to consider also values not in the image of X. That is, f_X may be defined for all real numbers and f_X(x) = 0 for all x $\notin$ X(S) as shown in the figure.

Since the image of X is countable, the probability mass function f_X(x) is zero for all but a countable number of values of x. The discontinuity of probability mass functions is related to the fact that the cumulative distribution function of a discrete random variable is also discontinuous. Where it is differentiable, the derivative is zero, just as the probability mass function is zero at all such points.

Examples

Suppose that S is the sample space of all outcomes of a single toss of a fair coin, and X is the random variable defined on S assigning 0 to "tails" and 1 to "heads". Since the coin is fair, the probability mass function is

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

This is a special case of the binomial distribution.

An example of a multivariate discrete distribution, and of its probability mass function, is provided by the multinomial distribution.

Further reading

• Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9 (p 36)