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| [[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.]]
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| In [[probability theory]] and [[statistics]], a '''probability mass function''' ('''pmf''') is a function that gives the probability that a [[discrete random variable|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=http://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.
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| 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.<ref>[http://mathworld.wolfram.com/ProbabilityFunction.html Probability Function] at Mathworld </ref>
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| ==Formal definition==
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| [[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.]]
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| Suppose that ''X'': ''S'' → ''A'' (A <math>\subseteq</math> '''R''') is a [[discrete random variable]] defined on a [[sample space]] ''S''. Then the probability mass function ''f''<sub>''X''</sub>: ''A'' → [0, 1] for ''X'' is defined as<ref>{{cite book|author=Kumar, Dinesh|title=Reliability & Six Sigma|publisher=Birkhäuser|year=2006|isbn=978-0-387-30255-3|page=22|url=http://books.google.com/books?id=XsX20uCFJbYC&pg=PA22}}</ref><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=http://books.google.com/books?id=nuoryE4IwMoC&pg=PA717}}</ref>
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| :<math>f_X(x) = \Pr(X = x) = \Pr(\{s \in S: X(s) = x\}).</math>
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| Thinking of probability as mass helps avoiding mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes ''x'':
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| :<math>\sum_{x\in A} f_X(x) = 1</math>
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| 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_(mathematics)|image]] of ''X''. That is, ''f''<sub>''X''</sub> may be defined for all [[real number]]s and ''f''<sub>''X''</sub>(''x'') = 0 for all ''x'' <math>\notin</math> ''X''(''S'') as shown in the figure.
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| Since the image of ''X'' is [[countable]], the probability mass function ''f''<sub>''X''</sub>(''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.{{Citation needed|date=April 2012}}
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| ==Examples==
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| 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
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| :<math>f_X(x) = \begin{cases}\frac{1}{2}, &x \in \{0, 1\},\\0, &x \notin \{0, 1\}.\end{cases}</math>
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| This is a special case of the [[binomial distribution]].
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| An example of a multivariate discrete distribution, and of its probability mass function, is provided by the [[multinomial distribution]].
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| ==References==
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| {{refimprove|date=April 2012}}
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| {{reflist}}
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| ==Further reading==
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| *Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9 (p 36)
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| {{Theory of probability distributions}}
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| [[Category:Probability theory]]
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| [[Category:Types of probability distributions]]
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