Probability and Statistics

Problem 6 Assume your computer only has the capacity to generate uniform random variables be- tween 0 and 1. Consider a "one-sided" normal random variable, i.e., with density function . e -12/2 for r > 0 and 0 otherwise. (a) Derive an acceptance-rejection method to generate a "one-sided" normal random variable using the exponential random variable with rate parameter 1 as the proposal. (b) Suppose you use the algorithm in part (a) to successfully generate the following output sequence: 0.20, 1.20, 1.13, 0.30, 1.23, 1.39, 1.24, 0.36, 1.21, 0.95, 1.18, 0.66, 0.50, 0.28, 0.05, 1.19, 1.37, 0.25, 0.47, 1.29, 1.08, 0.50, 1.17, 0.33, 0.05, 0.14, 0.41, 0.76, 0.12, 1.75. Use these to give a point estimate and a 95% confidence interval of P(0.5