Editing Event (probability theory) (section)

==A note on notation==

Even though events are subsets of some sample space <math>\Omega,</math> they are often written as predicates or indicators involving [[random variable]]s. For example, if <math>X</math> is a real-valued random variable defined on the sample space <math>\Omega,</math> the event
<math display=block>\{ \omega \in \Omega \mid u < X(\omega) \leq v \}\,</math>
can be written more conveniently as, simply,
<math display=block>u < X \leq v\,.</math>
This is especially common in formulas for a [[probability]], such as
<math display=block>\Pr(u < X \leq v) = F(v) - F(u)\,.</math>
The [[Set (mathematics)|set]] <math>u < X \leq v</math> is an example of an [[inverse image]] under the [[Map (mathematics)|mapping]] <math>X</math> because <math>\omega \in X^{-1}((u, v])</math> if and only if <math>u < X(\omega) \leq v.</math>