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In [[probability theory]], a '''probability mass function''' (abbreviated '''pmf''') gives the probability that a [[discrete random variable|discrete]] [[random variable]] is exactly equal to some value. The probability mass function differs from the [[probability density function]] in that the latter, defined only for [[continuous random variable]]s, does not describe an actual probability but rather a rate of change in the [[cumulative distribution function]].
==Mathematical description==
Suppose that ''X'' is a discrete random variable, that is, that it takes values on some [[countable]] [[
:<math>f_X(x) = \begin{cases}\mathrm{Pr}(X = x), &x\in S,\\0, &x\in \mathbb{R}\backslash S.\end{cases}</math>
Note that this explicitly defines ''f''<sub>''X''</sub>(''x'') for all [[real number]]s, including all values in '''R''' that ''X'' could never take; indeed, it assigns such values a probability of zero. (Alternatively, think of Pr(''X'' = ''x'') as 0 when ''x'' ∈ '''R'''\''S''.)
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