Definition:Random Variable/Continuous

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$ such that the domain of $X$ is a continuum.

We say that $X$ is a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$ if and only if:

$\ds \lim_{\delta x \mathop \to 0} \map \Pr {X \in \openint x {x + \delta x} } = \map f x \delta x$

where $f$ is the frequency function on $X$.

Also known as

Other words used to mean the same thing as random variable are:

stochastic variable

chance variable

variate.

The image $\Img X$ of $X$ is often denoted $\Omega_X$.

Also see

• Definition:Absolutely Continuous Random Variable
• Definition:Singular Random Variable

• Results about continuous random variables can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): continuous random variable
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): random variable
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continuous random variable
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): random variable
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): continuous random variable