University Statistics Assignment: Importance Sampling, Markov Chains

This statistics assignment delves into several key concepts including importance sampling, Markov chains, and Monte Carlo methods. Problem 1 explores the properties of instrumental distributions and their impact on variance reduction. Problem 2 involves sampling from a normal distribution using importance sampling and comparing the results to direct sampling, including the calculation of weighted mean and variance. Problem 3 focuses on Markov chains, specifically doubly stochastic matrices and their limiting state probabilities, as well as the transition probabilities. Problem 4 uses R programming to implement and analyze the Metropolis-Hastings algorithm for sampling from a target distribution. The document contains the solutions to each problem and demonstrates the application of these statistical methods through both theoretical analysis and practical R code implementations. The solution explores topics like variance reduction techniques and the application of Markov Chain Monte Carlo methods for integration and sampling.