# Continuous Random Variable

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

## 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, $\displaystyle{ (\Omega, \mathcal{F}, P) }$ is the probability space. For real observation space, the function $\displaystyle{ X\colon \Omega \rightarrow \mathbb{R} }$ is a real-valued random variable if :$\displaystyle{ \{ \omega : X(\omega) \le r \} \in \mathcal{F} \qquad \forall r \in \mathbb{R}. }$ This definition is a special case of the above because the set $\displaystyle{ \{(-\infty, r]: r \in \R\} }$ 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 $\displaystyle{ \{ \omega : X(\omega) \le r \} = X^{-1}((-\infty, r]) }$.

### 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.

### 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 $\displaystyle{ X }$, 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 $\displaystyle{ Y }$ is also a probability density function, fY(y), but fY(y) in this case is not the probability that the random variable $\displaystyle{ Y }$ takes on the value y. Rather, fY(y) is a continuous curve having the property that for all $\displaystyle{ a }$ and $\displaystyle{ b }$,
• P(a$\displaystyle{ Y }$b) = P({s(∈)S| $\displaystyle{ a }$Y(s) ≤ b}) = Integral(a,b). “fY(y) dy]