Conditional expectation

Definition

Let $X$ be an integrable real random variable on a probability space $(\Omega, \mathcal{F}, P)$, and let $\mathcal{G} \subseteq \mathcal{F}$ be a σ-algebra. A conditional expectation of $X$ given $\mathcal{G}$ is an integrable random variable $Z$ with these two properties:

1. $Z$ is $\mathcal{G}$-measurable.
2. $E[Z \, 1_G] = E[X \, 1_G]$ for each $G \in \mathcal{G}$. Such a $Z$ exists, and two such random variables are equal almost surely. One writes $Z = E[X \mid \mathcal{G}]$. For a random variable $Y$, $E[X \mid Y] = E[X \mid \sigma(Y)]$, where $\sigma(Y)$ is the σ-algebra generated by $Y$.

Remarks

The existence follows from the Radon–Nikodym theorem. For $E[X^2] < \infty$, $E[X \mid \mathcal{G}]$ is the orthogonal projection of $X$ onto $L^2(\mathcal{G})$. For a discrete $Y$, $E[X \mid Y] = g(Y)$ with $g(y) = E[X \, 1_{\{Y = y\}}] / P(Y = y)$ when $P(Y = y) > 0$. This $g(y)$ is the expectation of $X$ under the conditional probability $P(\cdot \mid Y = y)$. The conditional expectation is linear and monotone. If $W$ is $\mathcal{G}$-measurable and $WX$ is integrable, then $E[WX \mid \mathcal{G}] = W \, E[X \mid \mathcal{G}]$. If $X$ is independent of $\mathcal{G}$, then $E[X \mid \mathcal{G}] = E[X]$.

Explanations

L1 human

$E[X \mid Y]$ is the average value of $X$ when you know $Y$. It changes with $Y$, so it is a random variable. For two dice with results $D_1$ and $D_2$, $E[D_1 + D_2 \mid D_1] = D_1 + 3.5$.