Derive distribution of an inverse transform
Company: Citadel
Role: Data Scientist
Category: Statistics & Math
Difficulty: medium
Interview Round: Technical Screen
Let X have density f_X on (0,1), and define the strictly increasing logistic map g(u) = 1/(1 + e^{-u}). (A) Derive the general formula for the density of U = g^{-1}(X) in terms of f_X and g. (B) Take X ~ Uniform(0,1) and compute the distribution, mean, and variance of U = log(X/(1−X)). Clearly show the Jacobian steps and support.
Quick Answer: This question evaluates change-of-variables and inverse-transform techniques in probability, focusing on the logit (inverse logistic) mapping, Jacobian-based density transformation, handling of support, and computation of distributional moments.