![]() (PART C HAS BEEN SOLVED) (d) Show that B and Y bar are independent of each other and of all the residuals R_i for i=1,2,…,n. It follows from (c) that we have an unbiased estimator of A + Bx_i of the linear component of α+βx_i that appears in the expression for Y_i, The remaining part R_i=Y_i-A-Bx_i is called the ith residual. (PART B HAS BEEN SOLVED) (c) Deduce that B = S_xY/S_xx is an unbiased estimator of β and A=Ybar - Bxbar is an unbiased estimator of α. (PART A HAS BEEN SOLVED) (b) Derive the distribution of S_xY=∑_(n, i=1)((x_i- x bar)(Y_i - Ybar)) in terms of α, β and S_xx=∑_(n, i=1) (x_i- xbar )^2. (a) Derive the distribution of Ybar=1/n ∑(n, i=1) (x_i - xbar) (Y_i - Ybar) in terms of α and β and x bar=1/n ∑_(n, i=1)^x_i. Note that x_i and Y_i are the only parts of this expression that are directly observed. Instead, we assume that there are unknown parameters for the intercept α and β in a linear equation y=α+βx, and we observe the values Y_i=α+βx_i+U_i,i=1,2,…,n Where x_i,…,x_n are known constants. In this problem, we suppose that the random variables U_1,U_2,…,U_n are not directly observed. Let U_1,U_2,…,U_n be independent and identically distributed normal random variables with expectation zero and unknown variance σ^2.
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