Problem 37 - A First Course in Probability

Question

A die is rolled twice. Let \(X\) equal the sum of the outcomes, and let \(Y\) equal the first outcome minus the second. Compute \(\operatorname{Cov}(X, Y)\)

Step-by-Step Solution

Answer

The covariance between X and Y is \(-7\).

Step 1: Compute E(X) and E(Y)

First, we need to find the expected values of X and Y. Since both A and B are independent and uniformly distributed over {1, 2, 3, 4, 5, 6}, we can use the properties of expectations to compute E(X) and E(Y): E(X) = E(A + B) = E(A) + E(B) = \(\frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6)\) + \(\frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6)\) = 7 E(Y) = E(A - B) = E(A) - E(B) = \(\frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6)\) - \(\frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6)\) = 0

Step 2: Compute E(X*Y)

Next, we need to find the expected value of the product of X and Y. Since X = A+B and Y = A-B, their product is: X*Y = (A + B)*(A - B) = \(A^{2}\) - \(B^{2}\) To find the expected value of X*Y, we can compute the sum of all possible values of \(A^{2}\) - \(B^{2}\) divided by the total number of possible outcomes (i.e., 36): E(X*Y) = \(\frac{1}{36}\) (sum of all possible values of \(A^{2}\) - \(B^{2}\)) To compute the sum, we can use a nested loop to iterate through all possible combinations of A and B: sum = 0 for A in {1, 2, 3, 4, 5, 6}: for B in {1, 2, 3, 4, 5, 6}: sum += A*A - B*B E(X*Y) = \(\frac{1}{36}\) * sum = -7

Step 3: Compute Cov(X, Y)

Finally, we can use the formula for covariance to compute Cov(X, Y): Cov(X, Y) = E(X*Y) - E(X)*E(Y) = -7 - 7*0 = -7 Therefore, the covariance between X and Y is -7.