A fair die is rolled 2 times.

A fair die is rolled 2 times. Let X1 and X2 denote the number of points showing on the first and second rolls, respectively. Let U=X1+X2 and V=X1-X2. (a)Find the mean and variance of V. (b)Find the covariance of U and V. (c)Show that U and V are not independent 

Here X1 and X2 denote the number of points showing on the first and second rolls so X1 and X2 are independent and both follows a discrete uniform distribution.

So,

X1~Discrete Uniform(1,6)

X2~ Discrete Uniform(1,6)

Thus,

E(X1) = E(X2) = 3.5 and V(X1) = V(X2) = .

a) E(V) = E(X1-X2) = E(X1) – E(X2) = 0

Var(V) = Var(X1-X2) = Var(X1) +V(X2) – 2Cov(X1,X2) = Var(X1) +V(X2)

= ; (covariance term is 0 as X1 and X2 are independent).

b) Cov(U, V) = Cov(X1+X2, X1-X2) = V(X1) + Cov(X2,X1) –Cov(X1,X2) –V(X2)

= 0

c) U and V are independent.

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