2. Best Linear Predictor of a Constrained Outcome Space Suppose that discrete random variables X and Y have joint probability mass function given by: 1/2: (2312;) E {(010)}(2:1)} 0, otherwise f($,y)={ (This means that there is equal probability that the points (0, ) and (2, 1) are drawn; there is zero probability that any other point is drawn.) Let 9(3) : g + ,8193 be a predictor for 3; that is a function of 3:, and dene the error, E, to be the difference between the true value of y and the prediction 9(a), E = Y 9(X). 1. (1 point) If you impose the moment condition (that is, you require that), E[E] = 0, what one point in the plane must the predictor pass through? (In some places, this point is referred to as the gmaal mean.) 2. (2 points) Because we have dened E = Y 9(X), we can ask the question, \" What 3's the covariance between X and E?\" Because how how we have dened E, we can know that the answer probably starts with a substitution: Coax, e] = Cover, Y gen Assume (or you might say, \"require\") that the expected value E is zero, E[E] = 0. Then, prove that cou[X, E] has the form a + 6,31. Given the constraints of the pdf, f (93,19), you have been provided, what is the specic value of b? 3. (2 points) How is the sign of cov[X, E] is related to the angle of the line. 4. (2 points) Compute the BLP in this way: a. Assume (or you might say require) that E[E] : D. b. Then, set 6'1)va , E] = 0 and solve for ,8]. \"'hat is the value of ,81