# Continuous Random Variable

(Redirected from real-valued random variable)

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A Continuous Random Variable ([math]\displaystyle{ X }[/math]) is a random variable that represents a continuous random experiment whose function range is an uncountable interval.

**Context:**- It can be associated with a Probability Density Function.
- It can be denoted by </math>X(x,y)</math> for a specific Function Range.
- It can can assume any value within one or more Real Intervals.
- It can range from being a Univariate Continuous Random Variable to being a Multivariate Continuous Random Variable (such as a bivariate continuous random variable).

**Example(s):**- [math]\displaystyle{ X(3,4) }[/math] ⇒ π.
- a Gaussian Random Variable.

**Counter-Example(s):****See:**Continuous Probability Space, Cumulative Density Function, Moment-Generating Function.

## References

### 2013

- (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Random_variable#Real-valued_random_variables
- In this case the observation space is the 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 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].

- http://en.wikipedia.org/wiki/Event_%28probability_theory%29#A_note_on_notation
- 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 :[math]\displaystyle{ \{\omega\in\Omega \mid u \lt X(\omega) \leq v\}\, }[/math] can be written more conveniently as, simply, :[math]\displaystyle{ u \lt X \leq v\,. }[/math] This is especially common in formulas for a probability, such as: [math]\displaystyle{ P(u \lt X \leq v) = F(v)-F(u)\,. }[/math] The set u*<*X ≤*v*is an example of an inverse image under the mapping*X*because [math]\displaystyle{ \omega \in X^{-1}((u, v]) }[/math] if and only if [math]\displaystyle{ u \lt X(\omega) \leq v }[/math].

- Even though events are subsets of some sample space Ω, they are often written as propositional formulas involving random variables. For example, if

### 2009

- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Random_variable#Formal_definition
- Let (Ω, \mathcal{F}, P) be a probability space and (\mathcal{}Y, \Sigma) be a measurable space. Then a random variable X is formally defined as a measurable function X: \Omega \rightarrow Y. An interpretation of this is that the preimages of the "well-behaved" subsets of Y (the elements of Σ) are events (elements of \mathcal{F}), and hence are assigned a probability by P.

### 2006

- (Dubnicka, 2006c) ⇒ Suzanne R. Dubnicka. (2006). “Random Variables - STAT 510: Handout 3." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
- TERMINOLOGY : A random variable is said to be continuous if its support set is uncountable (i.e., the random variable can assume an uncountably infinite number of values).
- ALTERNATE DEFINITION: A random variable is said to be continuous if its cdf FX(x) is a continuous function of x.
- TERMINOLOGY : Let X be a continuous random variable with cdf FX(x). The probability density function (pdf) for X, denoted by fX(x), is given by fX(x) = d/dx FX(x),

### 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*Y_{F}and*Y*. So the expected cost is:_{H}*E(C)*= …

- The cost, [math]\displaystyle{ C }[/math], is a '

### 1986

- (Larsen & Marx, 1986) ⇒ Richard J. Larsen, and Morris L. Marx. (1986). “An Introduction to Mathematical Statistics and Its Applications, 2nd edition." Prentice Hall
- '
*Definition 3.2.1. A real-valued function whose domain is the sample space S is called a*random variable*. We denote random variables by uppercase letters, often [math]\displaystyle{ X }[/math],*Y, or*Z*. - If the range of the mapping contains either a finite or countably infinite number of values, the random variable is said to be
*discrete*; if the range includes an interval of real numbers, bounded or unbounded, the random variable is said to be*continuous*. - …
- Associated with each continuous random variable [math]\displaystyle{ Y }[/math] is also a probability density function,
*f*(_{Y}*y*), but*f*(_{Y}*y*) in this case is*not*the probability that the random variable [math]\displaystyle{ Y }[/math] takes on the value*y*. Rather,*f*(_{Y}*y*) is a continuous curve having the property that for all [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math],*P*(*a*≤ [math]\displaystyle{ Y }[/math] ≤*b*) =*P*({*s*(∈)*S*| [math]\displaystyle{ a }[/math] ≤*Y*(*s*) ≤*b*}) = Integral(a,b). “f_{Y}*(*y*)*dy*]*

- '