Simulate via inverse transform and Gibbs

Overview: This question evaluates understanding of random variate generation and Markov chain Monte Carlo techniques, specifically inverse-transform sampling for the Logistic(μ,s) distribution and Gibbs sampling for a bivariate joint density, testing competency in probability theory, distribution properties, and statistical computing within the Statistics & Math domain. It is commonly asked to assess the ability to derive inverse CDFs and full conditional distributions, verify sample correctness with goodness-of-fit and convergence diagnostics, and reason about sample size, burn-in, and thinning, reflecting a mix of conceptual understanding and practical application.