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MisterSheik (talk | contribs) →Mathematical description: but if this was mathematically okay then we would just define it like that-- it would be much simpler. |
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Suppose that ''X'' is a discrete random variable, taking values on some [[countable]] [[sample space]] ''S'' ⊆ '''R'''. Then the probability mass function ''f''<sub>''X''</sub>(''x'') for ''X'' is given by
:<math>f_X(x) = \begin{cases} \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.
The discontinuity of probability mass functions reflects the fact that the cumulative distribution function of a discrete random variable is also discontinuous. Where it is differentiable (i.e. where ''x'' ∈ '''R'''\''S'') the derivative is zero, just as the probability mass function is zero at all such points.
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