Gaussian Share of a Sum

Key idea. Subtract from $X_1$ the multiple of the sum that is uncorrelated with the sum. For jointly normal variables, uncorrelated means independent.

Let $S=X_1+X_2$. The pair $(X_1,S)$ is a linear image of the independent normals $(X_1,X_2)$, so it is jointly normal. We have

Put $R=X_1-\tfrac{9}{25}S$. Then

Since $(R,S)$ is jointly normal and uncorrelated, $R$ is independent of $S$. Also $\mathbb{E}[R]=0$. Therefore

So each summand takes a share of the sum proportional to its variance.

At $S=5$ this gives $\tfrac{9}{25}\cdot 5=\tfrac95$.

The conditional variance is $\operatorname{Var}(R)=9-\tfrac{81}{25}=\tfrac{144}{25}$. Hence