Editing Probability mass function (section)

==Measure theoretic formulation==
A probability mass function of a discrete random variable <math>X</math> can be seen as a special case of two more general measure theoretic constructions: 
the [[probability distribution|distribution]] of <math>X</math> and the [[probability density function]] of <math>X</math> with respect to the [[counting measure]].  We make this more precise below.

Suppose that <math>(A, \mathcal A, P)</math> is a [[probability space]]
and that <math>(B, \mathcal B)</math> is a measurable space whose underlying [[sigma algebra|σ-algebra]] is discrete, so in particular contains singleton sets of <math>B</math>. In this setting, a random variable <math> X \colon A \to B</math> is discrete provided its image is countable.
The [[pushforward measure]] <math>X_{*}(P)</math>—called the distribution of <math>X</math> in this context—is a probability measure on <math>B</math> whose restriction to singleton sets induces the probability mass function (as mentioned in the previous section) <math>f_X \colon B \to \mathbb R</math> since <math>f_X(b)=P( X^{-1}( b ))=P(X=b)</math> for each <math>b \in B</math>.

Now suppose that <math>(B, \mathcal B, \mu)</math> is a [[measure space]] equipped with the counting measure <math>\mu</math>.  The probability density function <math>f</math> of <math>X</math> with respect to the counting measure, if it exists, is the [[Radon–Nikodym derivative]] of the pushforward measure of <math>X</math> (with respect to the counting measure), so <math> f = d X_*P / d \mu</math> and <math>f</math> is a function from <math>B</math> to the non-negative reals.  As a consequence, for any <math>b \in B</math> we have
<math display="block">P(X=b)=P( X^{-1}( b) ) = X_*(P)(b) = \int_{ b } f d \mu = f(b),</math>

demonstrating that <math>f</math> is in fact a probability mass function.

When there is a natural order among the potential outcomes <math>x</math>, 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 <math>X</math>. That is, <math>f_X</math> may be defined for all [[real number]]s and <math>f_X(x)=0</math> for all <math>x \notin X(S)</math> as shown in the figure.

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 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>. [[Discretization of continuous features|Discretization]] is the process of converting a continuous random variable into a discrete one.