STAT 3006 Assignment 3 Due date: 5:00 pm on 26 April (30%)Q1: There are 100 samples {X1 , X2 , . . . , X100 }. For each sample i, Xi = (Xi1 , Xi2 ). You know that these samples are from three clusters. For sample i, we use Zi to denote the cluster number to which sample i belongs. The proportion of the three clusters is denoted by 1 , 2 , 3 . Specifically, for each sample i, P (Zi = k) = k , 1 k 3. Given Zi = k, Xi1 N (1k , 1) and Xi2 N (2k , 1). Use Q1 dataset to estimate parameters (1k , 2k ), k = 1, 2, 3, (1 , 2 , 3 ) and Zi (1 i 100). Note: assign Dirichlet(2, 2, 2) prior to (1 , 2 , 3 ), and assign uniform prior p(jk ) 1 to jk , j = 1, 2; k = 1, 2, 3; implement Gibbs sampler 10,000 iterations, and only samples in the last 8,000 iterations are kept; use posterior mean to estimate , and use posterior mode to estimate Z . (35%)Q2: There are T = 1, 500 independent normal distributed random variables {X1 , X2 , . . . , XT }. Xt (1 t T ) denotes the value we observed at time t. We know that there are two mean-shift change points in this data stream. More specifically, there are two different change points at time k and l (1