Continuous 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.



  • (Wikipedia, 2013) ⇒
    • 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].


  • (Wikipedia, 2009) ⇒
    • 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.




  • (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, fY(y), but fY(y) in this case is not the probability that the random variable [math]\displaystyle{ Y }[/math] takes on the value y. Rather, fY(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). “fY(y) dy]