Probability mass function: Difference between revisions

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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 is also discontinuous. If <math>X</math> 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 occurrences); 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]] <math>X</math>, for which <math>P(X = x) = 0</math> for any possible <math>x</math>: in fact, 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.

==Examples==