Maximize Weighted Sum of Correlated Normals

This question considers how to optimally combine two correlated normal random variables under a simple linear constraint on the weights. The setting can be viewed either as a statistical weighting problem or as a physical analogy involving noise or vibration levels from two sources. The candidate is asked to reason about how to choose a position or weight along a one-dimensional continuum so that the resulting combined variability is as small as possible, given that both sources are always present to some degree. Variants of this appear in risk minimization, basic portfolio selection, and signal processing, where one wants the "smoothest" or least volatile outcome from noisy inputs.

Solving it leans on linear algebra and multivariate normal theory: variances and covariances arranged as a quadratic form in the weights, reduction of dimension using the linear constraint, and calculus to identify and classify critical points. The interviewer is watching whether the candidate can translate a probabilistic description into an optimization problem, manipulate covariance terms correctly, and argue why the extremum found is indeed a minimum. They may also look for geometric intuition about correlated Gaussian vectors and how dependence changes the optimal combination.

What it tests

Whenever you are asked to minimize (or maximize) the variance of a linear combination of correlated random variables subject to a linear constraint on the weights, you are dealing with a quadratic optimization problem under linear constraints. The variance of a linear combination, say $S = aX + bY$, expands to a quadratic form in the weights, reflecting both the individual variances and the covariance structure. The constraint (like $a + b = 1$) allows you to reduce the dimensionality of the problem, expressing everything in terms of a single variable. The minimum of this quadratic function occurs at a critical point found by differentiation, and the nature of the quadratic ensures that, unless the variables are perfectly correlated, this critical point is a true minimum. This structure is fundamental in portfolio theory and risk minimization, where the interplay between variances and covariances determines the optimal mix.