Expectation of Sum of Normals

This quant interview question is about conditional expectation in the setting of continuous random variables, specifically within the multivariate normal world. You have independent Gaussian inputs and an observed linear combination, and you want the expected value of one component given that observation. It sits at the crossroads of probability theory and statistics, a sweet spot for serious quant prep and interviews at top trading firms.

It trains your understanding of how Gaussian vectors behave under linear constraints and how information about a sum reshapes your beliefs about a single component. You are being pushed to connect variance, correlation structure, and conditional distributions, and to manipulate expectations in a precise, quantitative way.

This matters for quant interviews because many pricing, risk, and signal models assume normality, linear factors, and conditional relationships. Interviewers use this to test whether you can reason about continuous random variables under conditioning, a core skill in high-level quant prep for trading, research, and risk roles.

What it tests

When dealing with the conditional expectation of one component of a sum of independent normal random variables, the key structure is that the conditional distribution remains normal, and the conditional mean is a weighted average of the observed sum, with weights determined by the variances. Specifically, for $X_1$ and $X_2$ independent normals, $\mathbb{E}[X_1 \mid X_1 + X_2 = s]$ is $\frac{\operatorname{Var}(X_1)}{\operatorname{Var}(X_1) + \operatorname{Var}(X_2)} s$. This arises because, under the constraint $X_1 + X_2 = s$, the most likely split of $s$ between $X_1$ and $X_2$ is proportional to their variances: the variable with larger variance is more likely to take on more extreme values. The underlying reason is that the joint normal density penalizes deviations from the mean quadratically, and the sum constraint links the two variables, so the conditional mean must balance these penalties according to the variances.