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Question 2. [18 marks] Consider the two-dimensional random vector (X, Y), where X is a continuous random variable and Y is a scalar binary-valued random variable. Assume the following conditional distribution of X given Y: X Y =i~ N(Mi, o?), i =0,1, (2) where the means and covariances are distinct for all i = 0, 1. The marginal distribution of Y is P (Y = 0) = 0 and P (Y = 1) = 1 -0 for some 0 E [0, 1]. (a).[6 marks ] Suppose we observe n i.i.d sample observations of this form, {(X,, Y;)}}=1. Find the minimal sufficient statistics for unknown parameters {(Mi, o?) }i=o,1 and 0. (b).[6 marks ] Derive an expression for the marginal distribution of X. The marginal distribution is also a function of { (Mi, o?) }izo,1 and 0. (c). [6 marks ] Suppose we can only observe X and not Y, and that we have observations {X,) =1 generated according to the marginal distribution derived in part (b). What are the minimal sufficient statistics for { (Mi, ?) }=] and 0 in this situation