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 $f=dX_{*}P/d\mu$ 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\}}fd\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

↑ 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.

Theory of probability distributions

probability mass function (pmf) probability density function (pdf) cumulative distribution function (cdf) quantile function

raw moment central moment mean variance standard deviation skewness kurtosis L-moment

moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant

Probability mass function

Formal definition

Measure theoretic formulation

Examples

References

Further reading