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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 is exactly equal to some value. 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.〔(Probability Function ) at Mathworld 〕 ==Formal definition== Suppose that ''X'': ''S'' → ''A'' (A R) is a discrete random variable defined on a sample space ''S''. Then the probability mass function ''f''''X'': ''A'' → (1 ) for ''X'' is defined as : 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'': : 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'' ''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「probability mass function」の詳細全文を読む スポンサード リンク
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