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 Cov(X, Y).

Answer 1

Cov(X, Y) = 0

Explanation:

X = Sum of the outcomes

Y = Difference between the outcomes (first - second)

Let the two outcomes of the die be A and B

X = A + B

Y = A - B

Cov(X, Y) = Cov( A+B, A-B)

Cov(X, Y) = Cov(A,A) - Cov(A,B) + Cov(B,A) - Cov(B,B)

Cov(X,Y) = Cov(A, A) - Cov(B, B)

Cov(X, Y) = Var(A) - Var(B)

A and B are identically distributed, Var(A) = Var(B)

This means that: Cov(X, Y) = 0