Real-Valued Random Variable
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A Real-Valued Random Variable is a Measurable Function which the outcome is a set of real numbers.
- AKA: Random Variable.
- Example(s):
- Counter-Example:
- See: Continuous Random Variable, E-Valued Random Variable.
References
2019
- (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Random_variable#Real-valued_random_variables Retrieved:2019-8-31.
- In this case the observation space is the set of real numbers. Recall, [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math] is the probability space. For real observation space, the function [math]\displaystyle{ X\colon \Omega \rightarrow \mathbb{R} }[/math] is a real-valued random variable if :
[math]\displaystyle{ \{ \omega : X(\omega) \le r \} \in \mathcal{F} \qquad \forall r \in \mathbb{R}. }[/math]
This definition is a special case of the above because the set [math]\displaystyle{ \{(-\infty, r]: r \in \R\} }[/math] generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that [math]\displaystyle{ \{ \omega : X(\omega) \le r \} = X^{-1}((-\infty, r]) }[/math] .
- In this case the observation space is the set of real numbers. Recall, [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math] is the probability space. For real observation space, the function [math]\displaystyle{ X\colon \Omega \rightarrow \mathbb{R} }[/math] is a real-valued random variable if :
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]\displaystyle{ C }[/math], is a 'real-valued random variable defined as the loss over the random variables YF and YH. So the expected cost is: E(C) = ...