1. Assume that for a given day, your number of phone call, denoted by n, has a Pois- son distribution with mean 1. Let yi..... Yn be the duration for each call. Given n, Y1, ..., yn are independently distributed with the Exponential(6) distribution. (a) Write out the joint likelihood function p(n, y1,..., yn|1,0) and identify the support for each parameter. (b) Consider the non-informative flat prior for the land o. Find the posterior distri- bution of land o and prove this is a proper distribution. (c) Suppose that today you have n calls with duration yi,... Yn. Let the number of phone call you will have tomorrow and y1, ... n the corresponding duration. Pick one problem from the following two options. 1. Derive the the predictive distribution p,1,..., ynn, y1,..., Yn) analytically. or 2. Describe in pseudo code how you would generate samples for the predictive distribution p(, yi,...,Un|n, y1, ..., yn). (d) Following part (c), given that you had 4 calls today with a total duration of 780 seconds, what is the expectation for the total duration of your phone calls tomorrow