233 X and Y are independent zero mean Gaussian random variab

2.33 X and Y are independent zero mean Gaussian random variables with variances sigma^2 x, and sigma^2 y. Let Z = 1/2(X + Y) and W =1/2 (X - Y) a. Find the joint pdf fz, w(z, w). b. Find the marginal pdf f(z). c. Are Z and W independent?

Solution

Since X and Y are independent (so that Cov(X, Y) = 0), the joint pdf if
fx,y(x, y) = (1/(2)) * exp[-x²/(2²) - y²/(2²)].

Letting Z = (1/2)(X + Y) and W = (1/2)(X - Y)
<==> X = Z + W and Y = Z - W.

(x,y)/(z,w) =
|1...1|
|1..-1| = -2.

So, fz,w (z,w) = (1/(2)) * exp[-(z+w)²/(2²) - (z-w)²/(2²)] * |-2|
.....................= (1/()) * exp[-(z+w)²/(2²) - (z-w)²/(2²)].