Correlation rho between two variables

Question

Roll a fair, six-sided die 12 times. Let X be the sum of the first 10 rolls. Let Y be the sum of the last 10 rolls. Find the correlation rho(X, T) between X and Y.

Answer 1

Let the variance of each roll be V

Then, var(X) = 10V, var(Y) = 10V, and cov(XY) = 8V, because there are 8 rolls in common - rolls 3-10

(To show explicitly, let x1 = r1+r2, x2=r3+r4+r5+r6+r7+r8+r9+r10, and y1 = r11+r12;

X = x1+x2; Y = x2 + y1

then, cov(X,Y) = cov(x1+x2,x2+y1) = cov(x1,x2) + cov(x1,y1)+cov(x2,x2)+cov(x2,y1) = 0 + 0 + 8V + 0 = 8V)

Then, ρ = cov(XY)/√(var(X)var(Y)) = 8V/10V = 4/5