Knowlpedia

Conditional expectation

An integrable random variable characterized by its averages over events in a given sigma-algebra.

A conditional expectation of an integrable XX given a sub-σ\sigma-algebra G⊆F\mathcal G\subseteq\mathcal F on a (Ω,F,P)(\Omega,\mathcal F,\mathbb P) is an integrable, G\mathcal G- YY such that

E ⁣[Y 1G]  =  E ⁣[X 1G]for every G∈G.\mathbb E\!\left[Y\,\mathbf 1_G\right] \;=\; \mathbb E\!\left[X\,\mathbf 1_G\right]\quad\text{for every }G\in\mathcal G.

Such a YY exists and is unique up to almost-sure equality; it is denoted E[X∣G]\mathbb E[X\mid\mathcal G].

Remarks

The special case X=1AX=\mathbf1_A gives the of AA given G\mathcal G.

Examples
  • If G={∅,Ω}\mathcal G=\{\varnothing,\Omega\}, then E[X∣G]=E[X]\mathbb E[X\mid\mathcal G]=\mathbb E[X] almost surely.
  • If XX is G\mathcal G-measurable, then E[X∣G]=X\mathbb E[X\mid\mathcal G]=X almost surely.
  • If B∈FB\in\mathcal F, P(B)∈(0,1)\mathbb P(B)\in(0,1), and G=σ(B)\mathcal G=\sigma(B), then
    E[X∣G]  =  E[X 1B]P(B) 1B  +  E[X 1Bc]P(Bc) 1Bc.\mathbb E[X\mid \mathcal G] \;=\; \frac{\mathbb E[X\,\mathbf 1_B]}{\mathbb P(B)}\,\mathbf 1_B \;+\; \frac{\mathbb E[X\,\mathbf 1_{B^c}]}{\mathbb P(B^c)}\,\mathbf 1_{B^c}.