Definition:Random Variable/Continuous/Singular
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Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.
Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.
We say that $X$ is singular if and only if:
- there exists a $\lambda$-null set $B$ such that $\map \Pr {X \in B} = 1$.
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
- Results about singular random variables can be found here.