Knowlpedia

Conditional expectation

Expectation of a random variable given partial information represented by a sigma-algebra

A conditional expectation of a XX given a sub- G⊆F\mathcal G \subseteq \mathcal F on a (Ω,F,P)(\Omega,\mathcal F,\mathbb P) is any G\mathcal G- YY such that XX is integrable (i.e. E[∣X∣]<∞\mathbb E[|X|]<\infty) and

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.

Any two versions of YY that satisfy this are equal almost surely, and one writes Y=E[X∣G]Y=\mathbb E[X\mid \mathcal G].

Remarks

Conditional expectation refines by restricting to information contained in G\mathcal G; the special case X=1AX=\mathbf 1_A yields of an event AA given G\mathcal G.

Examples
  • If G={∅,Ω}\mathcal G=\{\varnothing,\Omega\} is the trivial σ\sigma-algebra, then E[X∣G]=E[X]\mathbb E[X\mid\mathcal G]=\mathbb E[X] (a constant random variable).
  • If XX is G\mathcal G-measurable (in particular if G=F\mathcal G=\mathcal F), then E[X∣G]=X\mathbb E[X\mid\mathcal G]=X almost surely.
  • If B∈FB\in\mathcal F with P(B)∈(0,1)\mathbb P(B)\in(0,1) and G=σ(B)={∅,B,Bc,Ω}\mathcal G=\sigma(B)=\{\varnothing,B,B^c,\Omega\}, 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}.