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Analyze Correlations and Generate Gaussians

CitadelData Scientist · Technical Screen
MediumData ScientistTechnical ScreenMachine Learning

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This question evaluates understanding of correlation properties, feasible dependence constraints in multivariate distributions, construction of joint random variables with specified pairwise correlations, and methods for generating Gaussian samples from uniform random number generators, testing skills in probability, statistics, and simulation within the Machine Learning domain. It is commonly asked to assess both conceptual understanding of dependence structures and positive semidefiniteness and practical application of random variate generation techniques, reflecting a mix of conceptual and applied competencies for data scientist roles.

You are interviewing for a quantitative data science role. Answer the following probability and simulation questions:

  1. Let (X), (Y), and (Z) be random variables such that (\mathrm{Corr}(X,Y)=0.8) and (\mathrm{Corr}(X,Z)=0.9). Determine the full possible range of (\mathrm{Corr}(Y,Z)).
  2. As a follow-up, construct explicit random variables (X), (Y), and (Z) such that every pair has correlation (-1/2). Because correlation is symmetric, interpret the follow-up as asking for all three pairwise correlations to equal (-1/2).
  3. Suppose you only have access to a random number generator that returns independent samples from (\mathrm{Uniform}(0,1)). How can you generate samples from (\mathrm{Normal}(0,1))?