Sometimes it is known in advance that the least-squares regression line must go through the origin, i.e., the regression model is of the form Y=BX;+;. i=1.2. ... /. where &;'s are i.id. N(0, 6- ). and the equation of the regression line is y = B .x. In this case, finding the least-squares line reduces to finding the value B that minimizes the expression f(B)= [[y-B.x;]2. i-l Use the derivative of f with respect to B to derive the formula for the slope of the least-squares regression line in this case.\fSuppose you're given the following data X = (3, -1, 2), Y = (7,4,2). a) Fit the data to the linear regression model Yi = Bot Bite, i=1, ...,3. That is, compute the OLS estimates Bo and 8, for the parameters By and 8, respectively. b) Compute the fitted values Y, = Ao + AX; for i = 1, ... ,3. c) Compute the residuals c; = Y; - Y; for i = 1, ... ,3, and the residual sum of squares RSS d) Compute the (unadjusted) R? = 1 - RSS Tos of the regression. e) Compute the standard error of regression SER G. = \\ RSS. f) Compute the homoskedasticity-only standard error for 8, and the corresponding 95% CI. g) Compute the hetercoskedasticity-robust standard error for B, and the corresponding 95% CI. Which standard error is larger-the robust or non-robust one?3. Consider two binary random variables z and y having the joint distribution prescribed in the following table. Define entropy as H(X) = - Exp(X = x) Inp(X = x) (i.e. using nats); define the conditional entropy as H(Y|X) = - Ex E,p(X = x. Y = y) Inp(Y = y|X = x). Evaluate the following entropy related quantities 0 1/3 1/3 0 1/3 (a) H(x) (b) H(yz) (c) H(x, y)Consider two Bernoulli random variables, X and Y with joint pdf described as follows: fx.y(0, 0) = p, fx,y(1, 1) = p, fx,y(0, 1) = -. fx,y(1,0) = - - 2p, (a) Find values of p such that the joint distribution is correctly defined. (b) Describe E[Y|X] as a function of p. (c) Suppose p = . Show that we can express E[Y|X] = a + bX and find the values of th parameters a and b. (d) Find a value of p such that Cov(X, Y) = 0. (e) For the value of p you found above, are X and Y independent?Suppose you're given the following data X = (3, -1,2). Y = (7, 4,2). a) Fit the data to the linear regression model Y=Atta, i=1,...,3. That is, compute the OLS estimates jo and & for the parameters So and & respectively. b) Compute the fitted values Yi = 3p + PX for i = 1, ....3. c) Compute the residuals e = Y - Y, for { = 1, ... ,3, and the residual sum of squares RSS d) Compute the (unadjusted) R' = 1 _ RSS Tog of the regression. e) Compute the standard error of regression SER . = 1 7-7 1, RSS. f) Compute the homoskedasticity-only standard error for S, and the corresponding 95% CI. g) Compute the heteroskedasticity-robust standard error for 8, and the corresponding 95% CI. Which standard error is larger-the robust or non-robust one? h) Run the linear regression using R and inspect the output for standard error of 81. Which standard error does R report-the robust or non-robust one