1 Consider rolling two fair six sided dice (D1 & D2) and recording the sum of the faces and the maximum of the faces. Define two random variables Y = D1 + D2 and X = max{D1, D2}. The joint probability distribution P(X = x, Y = y) of these two random variables is provided in the lecture notes. Perform the following tasks: 1.i Compute the true conditional expectations: E[Y |X = x], x = 1, 2, 3, 4, 5, 6 1.ii Use R or Python to simulate n = 1000 draws of (X, Y ). Plot the n = 1000 draws. 1.iiii Assuming the conditional expectation takes on a linear form E[Y |X = x] = 0 + 1x, estimate 0 and 1 using ordinary least squares. Include this line on your scatter plot from Part 1.i. 1.iv Predict the true conditional expectations E[Y |X = x] using the estimated line 0+ 1x. Do you believe that these estimates are biased? 1.v Estimate the bias of Y (x) = 0 + 1x for estimating the true conditional expectation E[Y |X = x]. For example, the true bias at x = 3 is B(3) = E[Y