Cholesky for Correlated Normals

This question is about constructing correlated normal random variables from independent ones using a linear algebraic transformation. The setup focuses on a low-dimensional example: generating a pair of Gaussian variables with a desired correlation, starting from independent standard normals such as those produced by a basic random number generator. Candidates are expected to connect the target correlation structure to an appropriate covariance matrix, then interpret how a matrix factorization can turn independence into the required dependence. This style of question is common in quantitative finance and risk roles, where Monte Carlo simulation of correlated assets, factors, or risk drivers is routine.

The problem leans on understanding covariance matrices, positive semidefiniteness, and how Cholesky decomposition factors a covariance matrix into a product of a triangular matrix and its transpose. It tests the ability to translate a statistical requirement (given correlation) into a concrete matrix, carry out or describe the factorization, and then map that to a simple linear transformation of independent normals. Interviewers watch for comfort with linear algebra, clarity on why the resulting variables have the intended covariance, and avoidance of common misconceptions about correlation and dependence.

What it tests

Any multivariate normal distribution with a specified covariance structure can be constructed by applying a linear transformation to a vector of independent standard normal variables. The Cholesky decomposition provides a systematic way to find such a transformation: it factors the desired covariance matrix into a product of a lower triangular matrix and its transpose. By multiplying the vector of independent normals by this lower triangular matrix, the resulting vector has the desired covariances and variances. This works because linear combinations of jointly normal variables remain normal, and the covariance structure is preserved under linear transformations. The key is that the Cholesky factor encodes exactly how much each independent component should contribute to each correlated output.