Outcome Variable Expectation

Why

If we model some measured value as a random variable with induced distribution $p: V \to \R $, then one interpretation of $p(v)$ for $v \in V$ is the proportion of times in a large number of trials that we expect to measure the value $v$.¹

Definition

Given a distribution $p: \Omega \to \R $ and a real-valued outcome variable $x: \Omega \to \R $, the expectation (or mean) of $x$ under $p$ is $\sum_{\omega \in \Omega } p(\omega )x(\omega )$.

Notation

We denote the expectation of $x$ under $p$ by $\E _p(x)$. When there is no chance of ambiguity, we write $\E (x)$.

Properties

Let $x, y : \Omega \to \R $ be two outcome variables and $p: \Omega \to \R $ a distribution. Let $\alpha , \beta \in \R $. Define $z = \alpha x + \beta y$ by $z(\omega ) = \alpha x(\omega ) + \beta y(\omega )$. Then $\E (z) = \alpha \E (x) + \beta \E (z)$. Many authors refer to this property as the linearity of expecation.

Example: expectation

Suppose $\Omega = \set{1, 2, 3, 4, 5}$ with $p(1) = 0.1$, $p(2) = 0.15$, $p(3) = 0.1$, $p(5) = 0.25$ and $p(5) = 0.4$. Define $x: \Omega \to \R $ by

\[ x(a) = \begin{cases} -1 & \text{ if } a = 1 \text{ or } a = 2, \\ 1 & \text{ if } a = 3 \text{ or } a = 4, \\ 2 & \text{ if } a = 5. \\ \end{cases} \]

Two routes for computation

Denote by $p_x: V \to \R $ the induced distribution of $x: \Omega \to V$ (where $V \subset \R $). Then $\E (x) = \sum_{v \in V} p_x(v)v$ since

\[ \begin{aligned} \textstyle \sum_{\omega \in \Omega } p(\omega )x(\omega ) &= \sum_{v \in V} \sum_{\omega \in x^{-1}(v)} x(\omega )p(\omega ) \\ \textstyle &= \sum_{v \in V} v \sum_{\omega \in x^{-1}(v)} p(\omega ) \\ \textstyle & = \sum_{v \in V} x(v) p_x(v). \end{aligned} \]

Interpretations

We interpret the mean as the center of mass of the induced distribution.

1. Future editions may modify this explanation, and take a genetic approach via summary statistics. ↩︎