# Real-Valued Random Variable

**See:** Continuous 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).

### 1996

- (Kohavi & Wolpert, 1996) ⇒ Ron Kohavi, and David Wolpert. (1996). “Bias Plus Variance Decomposition for Zero-One Loss Functions.” In: Proceedings of the 13th International Conference on Machine Learning (ICML 1996).
- The cost, [math]C[/math], is a '
*real-valued random variable defined as the loss over the random variables*Y_{F}and*Y*. So the expected cost is:_{H}*E(C)*= ...

- The cost, [math]C[/math], is a '