4. Poisson Process MAP Customers arrive to a store according to a Poisson process of rate 1. The store manager learns of a rumor that one of the employees is sending 1/2 of the customers to the rival store. Refer to hypothesis X = 1 as the rumor being true, that one of the employees is sending every other customer arrival to the rival store and hypothesis X = 0 as the rumor being false, Where each hypothesis is equally likely. Assume that at time 0, there is a successful sale. After that, the manager observes 31, 5'2, . . . ,3\" where n is a positive integer and S,- is the time of the i th subsequent sale for i = 1,... ,n. Derive the MAP rule to determine Whether the rumor was true or not. 5. Gaussians and the MSE Suppose you draw n i.i.d. data points (x1, y1) , ..., (In, Un), where n is a positive integer and the true relationship is Y = WX +E,& ~ N (0, 02) . (That is, Y has a linear dependence on X, with additive Gaussian noise.) Show that finding the MLE estimate of W given the data points { (X;, yi) : i = 1, ..., n} is equivalent to minimizing the cost function n J(w) = (yi - wai)2 i=1