Editing Probability distribution (section)

===Indicator-function representation===
For a discrete random variable <math>X</math>, let <math>u_0, u_1, \dots</math> be the values it can take with non-zero probability. Denote
<math display="block">\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots</math>
These are [[disjoint set]]s, and for such sets
<math display="block">P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.</math>
It follows that the probability that <math>X</math> takes any value except for <math>u_0, u_1, \dots</math> is zero, and thus one can write <math>X</math> as
<math display="block">X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)</math>
except on a set of probability zero, where <math>1_A</math> is the indicator function of <math>A</math>. This may serve as an alternative definition of discrete random variables.