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Let = {a,b,c,d, e} be a sample space. Define two functions X and X on 2 by the following table a bcde 1 1 X(w) 0 0 1 X(W) 1 1 1 0 0 (a) Determine whether the following functions are o(X, X)-measurable. (i) Function U: R defined by (ii) Function V: R defined by (iii) Function Z: R defined by (b) We now suppose that there is a probability P({a}) = 1/8, P({b})= 1/8, P({c})= 1/4, U(a) = 1, V(a)= 1, Z(a)= 1, U(b) = 1, U(c) = 1, U(d) = 1, V(b) = 2, V(c)=3, V(d) = 4, Z(b) = 0, Z(c) = 1, Z(d) = 0, distribution P defined on 22 by U(e) = 1, V(e) = 5, Z(e) = 1. P({d})= 1/4, P({e})= 1/4. Find the best linear MMSE estimator of Z given X and X. (c) Continue using he probability distribution in (b), find the best nonlinear MMSE estimator of Z given X and X2.