Conditional mean of one normal given the sum

Let $X_2\sim N(0,16)$ and, independently of it, $X_1\sim N(0,9)$. You observe that their sum equals $5$. What is the conditional expectation of $X_1$?

$9/5$. For independent zero-mean normals, the sum is split in proportion to the variances: $\mathbb{E}[X_1\mid S=s]=\tfrac{\operatorname{Var}X_1}{\operatorname{Var}S}\,s=\tfrac{9}{25}\cdot5=\tfrac95$.