Outcome Variable Probabilities

Why

Given an outcome variable $X: \Omega \to V$, and some probabilities on a sample space $\Omega $, there is a natural set of probabilities to associate with outcomes in $V$.

Result

Suppose $X: \Omega \to V$ is a random variable on a finite sample space $\Omega $. Given $p: \Omega \to \R $ is a probability distribution inducing probability measure $P: \pow{\Omega } \to \R $, define the function $q: V \to \R $ by

\[ q(x) = P(X = x) \]

It is easy to verify that $q$ is nonnegative and normalized. The latter fact follows from the observation that the sets $\set{X^{-1}(x)}{x \in V}$ partition the set $\Omega $.

The function $q$ is sometimes called the distribution of $X$ (the induced distribution, induced probability distribution) of the random variable $X$.

Notation

Given a random variable $X: \Omega \to V$, and some distribution $p: \Omega \to \R $, it is common to denote the induced distribution by $p_X$. Given a distribution $q: V \to \R $ and a random variable $X: \Omega \to V$ it is common to see the notation $X \sim q$ as an abbreviation of the sentence “The random variable $X$ has distribution $q$.

Computation

Suppose $X: \Omega \to V$ is a random variable and $f: V \to U$. Define $Y: \Omega \to V$ by $y(\omega ) \equiv f(x(\omega ))$ for every $\omega \in \Omega $. We frequently denote this by $Y = f(X)$. $Y$ is a random variable with induced distribution $p_{Y}: \Omega \to \R $ satisfying

\[ \textstyle p_{Y}(y) = \sum_{\omega \in \Omega \mid Y(\omega ) = y} p(\omega ) = \sum_{x \in V \mid f(x) = b} p_X(x). \]

Consequently, as a matter of practical computation, we can evaluate probabilities having to do with the outcome variable $X$ using $p_X$ instead of $p$ and same with $Y$.¹

Future editions will give an example. ↩︎