Rutgers Business School Fall 2016 Introduction to Probability: Homework 9 Homework to be submitted on Blackboard by 10am on Wednesday November 23rd. Explain all your answers and show your working. Problem 1. Suppose a building project requires 20 tasks to be completed in succession. The time required for each task is an independent random variable with mean 5 days and standard deviation 1 day. A second building project also requires 20 tasks to be completed, and the time required for each task is an independent random variable with mean 5.2 days and standard deviation 1.5 days. (i) For each project, use Markov's Inequality and Chebyshev's inequality to give upper bounds on the probability that the project takes more than 110 days. (ii) For each project, use the Central Limit Theorem to estimate the probability that the project takes less than 90 days. (iii) Estimate the probability that the first project finishes before the second. Hint: Consider the random variable corresponding to the difference between the total time taken by the first project and the total time taken by the second. Problem 2. A die is repeatedly rolled until the total sum of the rolls is greater than 360. Estimate the probability that at least 80 rolls are necessary. 1