Real-Valued Random Variable

References

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Random_variable#Real-valued_random_variables
• Typically, the measurable space is the measurable space over the real numbers. In this case, let (\Omega, \mathcal{F}, P) be a probability space. Then, the function X: Ω \rightarrow \mathbb{R} is a real-valued random variable if
• { ω : X(ω) <= r } In F Forall r In R.
• This definition is a special case of the above because \{(-\infty, r]: r \in \R\} generates the Borel sigma-algebra on the real numbers, and it is enough to check measurability on a generating set. (Here we are using the fact that \{ ω : X(ω) \le r \} = X^{-1}((-\infty, r]).)
• http://en.wikipedia.org/wiki/Event_(probability_theory)
• Even though events are subsets of some sample space Ω, they are often written as propositional formulas involving random variables.

For example, if X is a real-valued random variable defined on the sample space Ω, the event

• ω | u < X(ω) <= v,
• It can be written more conveniently as, simply,
• u < X <= v.
• This is especially common in formulas for a probability, such as
• P(u < X <= v) = F(v)-F(u).