Real-Valued Random Variable

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See: Continuous Random Variable.


  • (Wikipedia, 2009) ⇒
    • 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]).)
    • 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).