University of Toronto Mississauga STA312- Topics in Statistics: Applied Statistical Modelling Assignment 1 Due date: Monday, October 3 Use R when that appropriate. Provide a print out of the results and the codes. Question 1 In a coin tossing experiment, let be the probability of heads. We observe 4 heads in 5 tosses of the coin. Suppose we use the prior of to be Beta(1/2, 1/2), find the posterior of . Plot the densities of both the prior and the posterior. Question 2 Let y1 , . . . , yn be an i.i.d sample from a geometric distribution with function p(yi |) = (1 )yi 1 , yi = 1, 2, . . . . It is assumed that the unknown parameter (0, 1) has prior Beta distribution, with given parameters > 0 and > 0. a. Find the posterior distribution of . b. Find the posterior mean. c. Find the posterior mode. d. Find the mle of . e. Show that the posterior mean converges to the mle, as n . Question 3 Let y1 , . . . , yn be an i.i.d sample from Exp() (i.e. p(yi |) = eyi , yi > 0). It is assumed that the unknown parameter > 0 has prior Gamma distribution, with given parameters > 0 and > 0. a. Find the posterior distribution of . b. Find the mean and variance of the posterior distribution of . c. Find the posterior mode. d. Find the posterior median. iid Question 4 Let y1 , . . . , yn N (, 1). Let the prior of be Beta(1, 1). a. Find the posterior of . b. Plot the prior and posterior densities on the same graph. c. Find the posterior mean and the posterior variance. d. Find the posterior predictive distribution. 1 e. Simulate a data set with n = 100 and = 3 (i.e. from N (3, 1)). Then simulate 1000 draws from the posterior. Now, plot a histogram of the simulated values from the posterior. In the same plot of the histogram, plot also the density of the posterior obtained part a. Compare the histogram to plot of the posterior density. iid Question 5 Let y1 , . . . , yn P oisson(). Suppose we use the prior Gamma(, ), where and are known positive numbers. a. Find the posterior distribution of . b. Show that the posterior expectation is a a convex combination of the prior mean and the sample average y. c. Find the posterior variance. d. Find the posterior predictive distribution. Question 6 Suppose that customers arrive at a restaurant during a time period of t hours follows with a Poisson distribution with mean t, where is the average hourly number of customers. Assume that the prior distribution for is p() 0.1 0.1 0.2 0.3 0.3 0.5 0.4 0.1 a. Compute the posterior distribution for if we observe 2 customers in 4 hours. b. Find the posterior mean c. What is the prior predictive distribution when there are 0 customers in 4 hours? 2