(2) Let Y be a linear function of X, i.e. Y- bo biX where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity we=E[rMb, that minimize«Q (a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ-ar 2 Cov2(x,Y) m Hint: You may use the fact that Q(bo,Y-YVar (Y -Y)+E (Y -Y)where Y.-bg + bİX...
Let Y be a linear function of X, i.e. Y = bo + bịX where bo and bl are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the values of bo and bi that minimizes Q (b) Use(a) to show that the minimal value of Q is Co - Hint: You may use the fact that Q(bg, b) [(Y-Y*)2] = Var (Y-Y*) +E(Y-Y*)]2 where Y*-bg + bİX and bi,...
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose that we would like to estimate the mean response at x = x*, that is we want to estimate lyx=* = Bo + B1 x*. The least squares estimator for /uyx* is = bo bi x*, where bo, b1 are the least squares estimators for Bo, Bi. ayx= (a) Show that the least squares estimator for...
(2) Let be a linear function of X, ie. = bo +b1X where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ -c 3xy 2 Cov2 (X.y Hint: You may use the fat that (b,bE[(Y -Yar (Y - Y)E(Y - Y) where Y.-bg +...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
Q3. The adder below adds two 16-bit numbers X and Y (i.e. S-X+Y), where X-Xi5Xi4...XiXo and Y-Y15Y14...YiYo. Assume we are using two's complement representation for our signed numbers, in which flipping all the bits of a number Y and adding one to it will give-Y. Modify the circuit below by including a signal P that picks whether the circuit will add them as X+Y, or subtract them as X-Y = X+(-Y) Suppose when P=0 the circuit will add and when...
3. Design a one bit subtractor. The circuit subtracts the number Y from X and generates a difference D bit and a borrow out Bo bit. The circuit has three inputs: X, Y and borrow in bit Bi shown in the figure below: во ID a. Derive the truth table for the function D(X,Y,Bi) and Bo(X,Y,Bi) b. Write the functions D and Bo as a canonical SOP form, using the little m notation. c. To implement the functions D and...
1. Let X and Y b e random variables, with μΧ = E(X), μΥ = E(Y), σ炙= Var(X) and σ Var(Y) (2) Let Ỹ be a linear function of X, ie. Ỹ = +51X where bo and bl are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ....
10.3.8 Suppose that Y = E(Y | X) + Z, where X, Y and Z are random variables. (a) Show that E (Z | X) = 0. (b) Show that Cov(E(Y | X), Ζ) = 0. (Hint. Write Z-Y-E(YİX) and use Theo- rems 3.5.2 and 3.5.4.) (c) Suppose that Z is independent of X. Show that this implies that the conditional distribution of Y given X depends on X only through its conditional mean. (Hint: Evaluate the conditional distribution function...
Problem 1: Consider the model Y = BO + Bi X+e, where e is a N(0,02) random variable independent of X. Let also Y = Bo + B1X. Show that E[(Y - EY)^3 = E[(Ỹ – EY)^3 + E[(Y – Y)1.