# Probability mass function

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.^{[1]} 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.^{[2]}

## Formal definition

Suppose that *X*: *S* → *A* (A **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^{[3]}^{[4]}

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

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* *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.

## Measure theoretic formulation

A probability mass function of a discrete random variable *X* can be seen as a special case of two more general measure theoretic constructions:
the distribution of *X* and the probability density function of *X* with respect to the counting measure. We make this more precise below.

Suppose that is a probability space
and that is a measurable space whose underlying σ-algebra is discrete, so in particular contains singleton sets of *B*. In this setting,
a random variable is discrete provided its image is countable.
The pushforward measure ---called a distribution of *X* in this context---is a probability measure
on *B* whose restriction to singleton sets induces a probability mass function since for each *b* in *B*.

Now suppose that is a measure space equipped with the counting measure μ. The probability density function *f* of *X* with respect to the counting measure, if it exists,
is the Radon-Nikodym derivative of the pushforward measure of *X* (with respect to the counting measure), so and *f* is a function from *B* to the non-negative reals. As a consequence, for any *b* in *B* we have

demonstrating that *f* is in fact a probability mass function.

## 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

This is a special case of the binomial distribution, the Bernoulli distribution.

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

## References

- ↑ Stewart, William J. (2011).
*Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling*. Princeton University Press. p. 105. ISBN 978-1-4008-3281-1. - ↑ Probability Function at Mathworld
- ↑ Kumar, Dinesh (2006).
*Reliability & Six Sigma*. Birkhäuser. p. 22. ISBN 978-0-387-30255-3. - ↑ Rao, S.S. (1996).
*Engineering optimization: theory and practice*. John Wiley & Sons. p. 717. ISBN 978-0-471-55034-1.

## Further reading

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