Problem 6. Suppose that X1, X2, . . . ,Xn are i.i.d. from the 2-Gaussian mixture model with unknown means and (known) unit variance. Furthermore, assume that the two components of the mixture have the same probability. In other words, there are unknown #1, n2 6 R, and the binary labels Y1,...,Yn are i.i.d. according to py = Berg). For each i = 1,...,n, independently generate Xe- according to the density function 1 _; $,_ 1 _1 $._ f($i|yi:6) = 2( ' \")2(1 2%) + 6 2( . \"mm (1) a M given Y; = yi, where 9 := (mum) is the parameter set. 0 What is the distribution of (Yh...,Yn) given the values of 6' and (X1,...,Xn) = ((171, . . . , III\")? 0 Suppose that only (XE-g, but not (Ya-g, are observed. Write out the update formula in the EM algorithm for estimating t9 (i.e., the formula for calculating 09\"\") based on 90%))