Question 3.1 a)

(3") Frequentist and Bayesian estimations [29 Marks] [ This question is a D level ( Distinction level) question. This question aims to demonstrate your expertise in the subject area). James, who drives to his workplace in the Melbourne city, often experiences difficulties in finding a vacant car parking spot near his workplace. He uses a mobile app that provides information about the parking availability in the city area along with its distance from his workplace. James wanted to estimate the probability of finding a vacant car parking spot that is within a particular short distance from his workplace. In order to estimate that, he collected data over several days by noting down the number of vacant car parking spots available, every day, using his mobile app. Assume that the number of vacant car parking slots available, denoted by x, on a day. i, follows a model that can be described using Poisson distribution with an unknown parameter, 0, that represents the probability of finding a vacant car parking spot, as given below: x ~ Poi(0) Poi(0) = p(x|0) = Xi! Assume that the number of vacant car parking spots available are independently and identically distributed (iid). 3.1) James first decided to use a frequentist approach to arrive at an estimate for 0. Answer the following questions. a) Show that the expression for the likelihood, p(X|0), of the number of vacant car parking spots available for N days (X = {X1, X2, ..., XN]) can be given by the below equation (show the steps clearly to obtain this). 0 e-NO = (olx)d X1! X2! X3!...XN! ' where S = Ext [3 marks]