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{{Short description|Discrete-variable probability distribution}}
{{refimprove|date=April 2012}}
[[Image:Discrete probability distrib.svg|right|thumb|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|probability]] 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.<ref>{{cite book|author=Stewart, William J.|title=Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling|publisher=Princeton University Press|year=2011|isbn=978-1-4008-3281-1|page=105|url=https://books.google.com/books?id=ZfRyBS1WbAQC&pg=PT105}}</ref>  The probability mass function is often the primary means of defining a [[discrete probability distribution]], and such functions exist for either [[Scalar variable|scalar]] or [[multivariate random variable]]s whose [[Domain of a function|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 probability density function are not probabilities as such: a PDF must be [[integration (mathematics)|integrated]] over an interval to yield a probability.<ref>[http://mathworld.wolfram.com/ProbabilityFunction.html Probability Function] at Mathworld</ref>

The value of the random variable having the largest probability mass is called the [[mode (statistics)|mode]].

==Formal definition==

Suppose that <math>X: S \mapsto A</math> for <math>A \subseteq \mathbb{R}</math> is a [[discrete random variable]] defined on a [[sample space]] <math>S</math>. Then the probability mass function <math>f_X: A \mapsto [0,1]</math> for <math>X</math> is defined as<ref>{{cite book|author=Rao, S.S.|title=Engineering optimization: theory and practice|publisher=John Wiley & Sons|year=1996|isbn=978-0-471-55034-1|page=717|url=https://books.google.com/books?id=nuoryE4IwMoC&pg=PA717}}</ref>
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|equation = <math>f_X(x) = P(X = x) = P(\{s \in S: X(s) = x\})</math>
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Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes <math>x</math>:
:<math>\sum_{x\in A} f_X(x) = 1</math>

When there is a natural order among the potential outcomes <math>x</math>, 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 (mathematics)|image]] of <math>X</math>. That is, <math>f_X</math> may be defined for all [[real number]]s and <math>f_X(x)=0</math> for all <math>x \notin X(S)</math> as shown in the figure.

The image of <math>X</math> has a [[countable]] subset on which the probability mass function <math>f_X(x)</math> is one. Consequently, the probability mass function is zero for all but a countable number of values of <math>x</math>.

The discontinuity of probability mass functions is related to the fact that the [[cumulative distribution function]] of a discrete random variable, when it is meaningful because there is a natural ordering, is also discontinuous.  Where it is differentiable, the derivative is zero, just as the probability mass function is zero at all such points.{{Citation needed|date=April 2012}} 
If ''x'' is a discrete random variable, then <math> P(X = x) = 1</math> means that the casual event <math>(X = x)</math> is certain (it is true in the 100% of the occurrencies); on the contrary, <math>P(X = x) = 0</math> means that the casual event <math>(X = x)</math> is always impossible. This statement isn't true for a [[continuous random variable]] ''X'', for which <math>P(X = x) = 0</math> for any possible ''x'': infact, by definition, a continuous random variable can have an [[infinite set]] of possible values and thus the probability it has a single particular value ''x'' is equal to <math>\frac{1}{\infty} = 0</math>. [[Discretization of continuous features|Discretization]] is the process of converting a continuous random variable into a discrete one.

==Measure theoretic formulation==

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

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

Now suppose that <math>(B, \mathcal B, \mu)</math> is a [[measure space]] equipped with the counting measure μ.  The probability density function <math>f</math> of <math>X</math> with respect to the counting measure, if it exists, is the [[Radon–Nikodym derivative]] of the pushforward measure of <math>X</math> (with respect to the counting measure), so <math> f = d X_*P / d \mu</math> and <math>f</math> is a function from <math>B</math> to the non-negative reals.  As a consequence, for any <math>b \in B</math> we have
:<math>P(X=b)=P(X^{-1}( \{ b \} )) := \int_{X^{-1}(\{b \})} dP =</math><math>\int_{ \{b \}} f d \mu = f(b),</math>
demonstrating that <math>f</math> is in fact a probability mass function.

==Examples==
===Finite===
[[Image:Fair dice probability distribution.svg|right|thumb|The probability mass function of a [[Dice|fair die]]. All the numbers on the {{dice}} have an equal chance of appearing on top when the die stops rolling.]]

*If the discrete distribution has two or more categories one of which may occur, whether or not these categories have a natural ordering, when there is only a single trial (draw) this is a [[categorical distribution]]. 

:Suppose that <math>S</math> is the sample space of all outcomes of a single toss of a fair coin, and <math>X</math> is the random variable defined on <math>S</math> assigning 0 to the category "tails" and 1 to the category "heads".  Since the coin is fair, the probability mass function is
::<math>f_X(x) = \begin{cases}\frac{1}{2}, &x \in \{0, 1\},\\0, &x \notin \{0, 1\}.\end{cases}</math>
:This is a special case of the [[binomial distribution]], the [[Bernoulli distribution]].

*An example of a [[Joint probability distribution|multivariate discrete distribution]], and of its probability mass function, is provided by the [[multinomial distribution]]. Here the multiple random variables are the numbers of successes in each of the categories after a given number of trials, and each non-zero probability mass gives the probability of a certain combination of numbers of successes in the various categories.

===Countably infinite===

*The following exponentially declining distribution is an example of a distribution with an infinite number of possible outcomes—all the positive integers:

::<math>\text{Pr}(X=i)= \frac{1}{2^i}\quad \text{for}\quad i=1, 2, 3, \dots .</math>

:Despite the infinite number of possible outcomes, the total probability mass is 1/2 + 1/4 + 1/8 + ... = 1, satisfying the unit total probability requirement for a probability distribution.

==Multivariate case==
{{Main|Joint probability distribution}}

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

==References==
{{reflist}}

==Further reading==
*{{cite book |last=Johnson |first=N. L. |last2=Kotz |first2=S. |last3=Kemp |first3=A. |year=1993 |title=Univariate Discrete Distributions |edition=2nd |publisher=Wiley |isbn=0-471-54897-9 |page=36 }}

{{Theory of probability distributions}}

[[Category:Types of probability distributions]]