Stat 244 Autumn 2021 — HW2 Due on Gradescope, Tuesday October 12 by 2pm 1. You have n random number generators, where the ith one draws a number uniformly at random from the interval [0, ti], for some constant ti 2 (0, 1). Let Xi be the number drawn by the ith random number generator. (a) What is the expected value of the sum, S = X1 + ··· + Xn? (Your answer will be in terms of t1,...,tn.) (b) What is the expected value of Y , which counts how many of the Xi’s are 1? (Your answer will be in terms of t1,...,tn.) 2. Let X ⇠ Uniform[0, 1]. (a) Calculate the density of the ratio R = X 1X . (b) Calculate the density of the product P = X(1 X). 3. Suppose that errors in HW solutions obey the following model. There is a 20% chance that there is an error in the posted solutions. If an error is present, then the number of students who send emails about the error, follows a Poisson(2) distribution. If there are no errors, then no emails are sent. If no emails were received, what is the probability that there was no error? 4. Calculate P(X is odd) in each setting below. Show your calculation or explain your answer for each part. (a) X ⇠ Geometric(0.7). Your final answer should be a number. (b) X ⇠ Binomial(101, 0.5). Your final answer should be a number. (c) First let Y ⇠ N(0, 1) (a standard normal random variable), and then let X be the answer you get when you round Y to the nearest integer. Your answer does not need to be simplified to a number—you can write it with summations/fractions/integrals etc as needed. However, your final answer cannot have any probability type notation in it, e.g. it cannot include terms like pX(3) or fY (1) or P(X 1) etc. 5. Let the random variable T be the time until some event occurs (e.g. time until an atom decays, time until next rainfall, etc). Suppose it’s a continuous random variable supported on [0, 1). The hazard rate for T is defined as h(t) = f(t) 1 F(t) , where f and F are the density and CDF for the distribution of T. On an intuitive level, this is the chance that the event will occur in the very near future, given that it has not yet occurred. Hazard rate is a function of time since it can rise or fall as time goes on. (a) Calculate the hazard rate h(t) if T ⇠ Exponential(). (b) Now suppose that T follows a Weibull distribution with shape k > 0 and scale ↵ > 0, which is supported on [0, 1) and defined by the CDF F(t)=1 e(t/↵)k on that interval. Calculate the density f(t), and the hazard rate function h(t), for this distribution. (c) For the Weibull distribution, for which values of k and ↵ is h(t) decreasing over time, increasing over time, or constant over t
