Probability mass function

In probability and statistics, a probability mass function (PMF), discrete probability density function, probability function, or frequency function is a function that gives the probability that a discrete random variable is exactly equal to some value. The PMF 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 PMF differs from a continuous probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A continuous PDF must be integrated over an interval to yield a probability.

The value of the random variable having the largest probability mass is called the mode.

Formal definition

A PMF is the probability distribution of a discrete random variable, and provides the possible values and their associated probabilities. It is the function ${\displaystyle p:\mathbb {R} \to [0,1]}$ defined by

for ${\displaystyle -\infty <x<\infty }$, where ${\displaystyle P}$ is a probability measure. ${\displaystyle p_{X}(x)}$ can also be simplified as ${\displaystyle p(x)}$.

The probabilities associated with all (hypothetical) values must be non-negative and sum up to 1,

$${\displaystyle \sum _{x}p_{X}(x)=1}$$ and $${\displaystyle p_{X}(x)\geq 0.}$$

Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes ${\displaystyle x}$.

Measure theoretic formulation

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

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

Now suppose that ${\displaystyle (B,{\mathcal {B}},\mu )}$ is a measure space equipped with the counting measure ${\displaystyle \mu }$. The probability density function ${\displaystyle f}$ of ${\displaystyle X}$ with respect to the counting measure, if it exists, is the Radon–Nikodym derivative of the pushforward measure of ${\displaystyle X}$ (with respect to the counting measure), so ${\displaystyle f=dX_{*}P/d\mu }$ and ${\displaystyle f}$ is a function from ${\displaystyle B}$ to the non-negative reals. As a consequence, for any ${\displaystyle b\in B}$ we have $${\displaystyle P(X=b)=P(X^{-1}(b))=X_{*}(P)(b)=\int _{b}fd\mu =f(b),}$$

demonstrating that ${\displaystyle f}$ is in fact a PMF.

When there is a natural order among the potential outcomes ${\displaystyle x}$, 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 ${\displaystyle X}$. That is, ${\displaystyle f_{X}}$ may be defined for all real numbers and ${\displaystyle f_{X}(x)=0}$ for all ${\displaystyle x\notin X(S)}$ as shown in the figure.

The image of ${\displaystyle X}$ has a countable subset on which the PMF ${\displaystyle f_{X}(x)}$ is one. Consequently, the PMF is zero for all but a countable number of values of ${\displaystyle x}$.

The discontinuity of PMFs is related to the fact that the cumulative distribution function of a discrete random variable is also discontinuous. If ${\displaystyle X}$ is a discrete random variable, then ${\displaystyle P(X=x)=1}$ means that the casual event ${\displaystyle (X=x)}$ is certain (it is true in 100% of the occurrences); on the contrary, ${\displaystyle P(X=x)=0}$ means that the casual event ${\displaystyle (X=x)}$ is always impossible. This statement isn't true for a continuous random variable ${\displaystyle X}$, for which ${\displaystyle P(X=x)=0}$ for any possible ${\displaystyle x}$. Discretization is the process of converting a continuous random variable into a discrete one.

Examples

Multivariate case

Two or more discrete random variables have a joint PMF, which gives the probability of each possible combination of realizations for the random variables.

References

Further reading

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