CS/DSC/AI 391L: Machine Learning Homework 4 - 'Theory Lecture: Prof. Qiang Liu 1. Assume X is a discrete random variable that takes values in {1, 2,3}, with probability defined by Pr(X = 1) =0, Pr(X = 2) = 20; Pr(X = 3) = 4s, where @ = [41,49] is an unknown parameter to be estimated. Now assume we observe a sequence D := fa), a), ol) that is independent and identi- cally distributed (i.i.d.) from the distribution. We assume the number of observations of the values: 1,2,3 in D are s1, S9, 53, respectively. (a) [5 points] To ensure that Pr(X = 7) is a valid probability mass function, what constraint should we put on @ = [@), 9]? Write your answers quantitatively as expressions that include 6) and @o. (b) [5 points] Write down the joint probability of the data sequence Pr(D | 0) =Pr({2,...,0(} |e), and the log probability log Pr(D | @). (c) [5 points] Calculate the maximum likelihood estimation 6 of @ based on the sequence D. 2. [10 points] Let fa), a al") } be an i.2.d. sample from an exponential distribution, whose the density function is defined as _ 1 = exp|], for O