Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions about maps from a probability space to a measure space which are measurable.

A random variable $X: \Omega \to E$ is a measurable function from a set of possible outcomes $\Omega$ to a measurable space $E$. The technical axiomatic definition requires $\Omega$ to be a sample space of a probability triple. Usually $X$ is real-valued.

The probability that $X$ takes on a value in a measurable set $S \subseteq E$ is written as :

$$Pr(X \in S) = P(\{ \omega \in \Omega|X(\omega) \in S\})$$

where $P$ is the probability measure equipped with $\Omega$.

history | excerpt history