Let X and Y be iid Unif0 1 Compute the covariance of X Y an

Let X and Y be i.i.d. Unif(0, 1). Compute the covariance of X + Y and X - Y. Are X + Y and X - Y independent?

Solution

Define U=X+Y, V=XY

Then, X=(U+V)/2, and

Y=(UV)/2.

Find the Jacobian J for the transformation.

Then, fU,V(u,v)= fX(x=(u+v)/2) fY(y=(uv)/2)|J|

You will find that fU,V(u,v) factors into a function of u alone and a function of v alone. Thus, by the Factorization thm, U and V are independent.

b) Yes X+Y and X-Y are independent

a) Since , X+Y and X-Y are independent,

COV(X+Y , X-Y) = 0 Answer

Cov(X+Y , X-Y) = 0

b)

Yes these are indipendent