Q3) Frequentist and Bayesian estimations [30 Marks] A software company designing marketing web pages, called SMkt Ltd, wants to learn about the average dwell time of a marketing web page, in order to help with their new marketing web page designs. The dwell time of a webpage is defined as the time a user spends on a web page. In order to model the dwell time, SMkt Ltd. assumes that the dwell time x; (in seconds) of a web page, i, follows an exponential distribution, p(x|2), with an unknown parameter (average dwell time), as given below. x~ p(x;\) = = 1 SMkt Ltd is interested in finding an estimate for the parameter . SMkt Ltd conducted a survey and collected dwell time data from N web pages. Assume that each of the web page dwell time is independently and identically distributed (iid). 3.1) SMkt Ltd first decided to use a frequentist approach to arrive at an estimate for . Answer the following questions. a) Show that the joint distribution of dwell times of N web pages can be given by the below equation (show the steps clearly to obtain this). P(X |) = where K = 1xi [3 marks] b) Find a simplified expression for the log-likelihood function L() = In (p(X|)) [3 marks] c) Show that the Maximum likelihood Estimate (^) of the parameter is given by: 1 N [4 Marks] = Xi N i=1 d) From the past history, the software company has found out that the dwell times of six of the web pages were {10, 12, 15, 20, 15, 20}. Find the Maximum likelihood Estimate of the parameter given this data, i.e., the estimate of the average dwell time? [2 Marks] e) Hence, find the probability that the dwell time of a web page lasts for more than 20 seconds. (Hint: you may use cumulative distribution function (cdf) of exponential - distribution. The cdf of exponential distribution is given by F(t) = 1 e ). [4 marks]