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.^[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

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^[3][4]

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

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:

$\sum_{x\in A} f_X(x) = 1$

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.

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 $(A, \mathcal A, P)$ is a probability space and that $(B, \mathcal B)$ is a measurable space whose underlying σ-algebra is discrete, so in particular contains singleton sets of B. In this setting, a random variable $X \colon A \to B$ is discrete provided its image is countable. The pushforward measure $X_*(P)$---called a distribution of X in this context---is a probability measure on B whose restriction to singleton sets induces a probability mass function $f_X \colon B \to \mathbb R$ since $f_X(b)=P(X^{-1}(b))=[X_*(P)](\{b\})$ for each b in B.

Now suppose that $(B, \mathcal B, \mu)$ 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 $f = d X_*P / d \mu$ and f is a function from B to the non-negative reals. As a consequence, for any b in B we have

$P(X=b)=P(X^{-1}( \{ b \} )) := \int_{X^{-1}(\{b \})} dP =$$\int_{ \{b \}} f d \mu = f(b),$

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

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

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

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

References

1. 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.
2. Probability Function at Mathworld
3. Kumar, Dinesh (2006). Reliability & Six Sigma. Birkhäuser. p. 22. ISBN 978-0-387-30255-3.
4. 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)