# 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 $Y$.