# Question: Suppose two random variables and are both zero mean and

Suppose two random variables and are both zero mean and unit variance. Furthermore, assume they have a correlation coefficient of ρ. Two new random variables are formed according to:

W = aX + bY,

Z = cX + dY,

Determine under what conditions on the constants a, b, c, and d the random variables W and Z are uncorrelated.

W = aX + bY,

Z = cX + dY,

Determine under what conditions on the constants a, b, c, and d the random variables W and Z are uncorrelated.

**View Solution:**## Answer to relevant Questions

Find an example (other than the one given in Example 5.15) of two random variables that are uncorrelated but not independent. Starting from the general form of the joint Gaussian PDF in Equation (5.40) and using the results of Exercise 5.35, show that conditioned on Y = y, X is Gaussian with a mean of μx + ρXY (σX / σY) (y – μY) and a ...A pair of random variables has a joint characteristic function given by (a) Find E [X] and E [Y] (b) Find E [XY] and Cov (X, Y). (c) Find E [X2Y2] and E [XY3]. For the joint CDF that is the product of two marginal CDFs, Fx,y (x, y) = Fx (x) Fy, as described in Exercise 5.4, show that the events {a< X < b}and {c < Y < d} are always independent for any constants a < b and c < d. Suppose X and Y are independent and Gaussian with means of μX and μY, respectively, and equal variances of σ2. The polar variables are formed according to R =√ X2 + Y2 and θ = tan–1 (Y / X). - Find the joint PDF of ...Post your question