Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 1

For an integrable random variable $X$ and a sub-sigma-algebra $\mathcal{G}\subseteq \mathcal{F}$, the conditional expectation $Y=\mathbb{E}[X\mid \mathcal{G}]$ is an integrable, $\mathcal{G}$-measurable random variable satisfyingfor every $G\in \mathcal{G}$.

Conditional expectation is unique up to almost sure equality. If $Y$ and $Z$ both satisfy the definition, then for every $G\in \mathcal{G}$,The events $\{Y>Z\}$ and $\{Z>Y\}$ belong to $\mathcal{G}$. Testing on them, or first on $\{Y-Z\ge 1/k\}$ and $\{Z-Y\ge 1/k\}$, shows that both have probability zero. Hence $Y=Z$ almost surely.

If $G\in \mathcal{G}\cap \mathcal{H}$, independence makes $G$ independent of itself, soThus every event in $\mathcal{G}\cap \mathcal{H}$ has probability zero or one. The intersection is trivial modulo null sets, and the independent sigma-algebras have trivial intersection result givesalmost surely.

Put $S=X_1+X_2$. By symmetry,Their sum is $S$, which is measurable with respect to $\mathcal{G}=\sigma (S)$, soThereforeThis also follows from the Gaussian conditional expectation formula because $\mathrm{Cov}(X_1,S)/\mathrm{Var}(S)=1/2$.

If $\mathcal{G}\subseteq \mathcal{H}$, then $\mathbb{E}[X\mid \mathcal{G}]$ is already $\mathcal{H}$-measurable, soIf $\mathcal{H}\subseteq \mathcal{G}$, the tower property of conditional expectation givesFinally, if $\mathcal{G}$ and $\mathcal{H}$ are independent, the $\mathcal{G}$-measurable variable $\mathbb{E}[X\mid \mathcal{G}]$ is independent of $\mathcal{H}$. Its conditional expectation given $\mathcal{H}$ is its mean $\mathbb{E}[X]$. Part c shows that the right side is also $\mathbb{E}[X]$.

The equation fails for general nonnested sigma-algebras. On the four-point space $\Omega =\{1,2,3,4\}$ with uniform probability, letand take $X=1_A$. The intersection $\mathcal{G}\cap \mathcal{H}$ is trivial, soBut $X$ is $\mathcal{G}$-measurable andwhich is zero on $B^c$ and is not almost surely $1/2$. This exhibits the failure of iterated conditional expectation over nonnested sigma-algebras.