Section 12.3

Conditioning on Sums and Symmetric Functions

E[X1 | X1+X2], Gaussian conditioning, E[X | X^2+Y^2] and similar, by symmetry or the regression formula.

Toolkit

When conditioning on a sum or a symmetric function of exchangeable or jointly Gaussian variables, exploit either (a) linearity plus symmetry, or (b) an explicit distributional symmetry (an involution preserving the conditioning variable) that pins down the conditional mean without computing densities.

Key tools:

Equal shares by symmetry/linearity: if $X_1,\dots,X_n$ are exchangeable and $S=X_1+\cdots+X_n$ , then $\mathbb{E}[X_i\mid S]=S/n$ for every $i$ .
Gaussian regression formula: for jointly normal $X,S$ ,
$\mathbb{E}[X\mid S]=\mathbb{E}[X]+\frac{\operatorname{Cov}(X,S)}{\operatorname{Var}(S)}\big(S-\mathbb{E}[S]\big).$
For a sum $S=\sum X_i$ of jointly normal variables with zero means, $\mathbb{E}[X_i\mid S]=\frac{\operatorname{Var}(X_i)+\sum_{j\ne i}\operatorname{Cov}(X_i,X_j)}{\operatorname{Var}(S)}S$ .
Involution trick: if $T=T(X)$ is a function invariant under some measure-preserving map $\phi$ (e.g. $X\mapsto -X$ , or a reflection $x\mapsto c-x$ ), then $\mathbb{E}[X\mid T]=\mathbb{E}[\phi(X)\mid T]$ ; solving this fixed-point equation (often forcing the mean to a symmetric point, or to $0$ ) gives the conditional expectation without integrating.

Problems