$ E[X] = E[Y] = 0 $

$ E[X] = \sigma_X^2 $

$ E[y] = \sigma_Y^2 $

cov[X,Y] = $ \rho * \sigma_X * \sigma_Y $

$ E[Z] = E[X cos \theta + Y sin \theta] = E[X]cos\theta + E[Y]sin\theta = 0 $

$ E[W] = E[Ycos\theta - X sin\theta] = 0 $

$ Var(Z) = E[Z^2] = E[X^2 cos^2\theta + Y^2 sin^2\theta + 2*\rho\sigma_x\sigma_y] $

$ Var(Z) = \sigma_X^2 cos^2\theta+\sigma_Y^2sin^2\theta+2*\rho\sigma_X\sigma_Ysin\theta *cos\theta $

$ Var(W) = \sigma_X^2 sin^2\theta+\sigma_Y^2cos^2\theta-2*\rho\sigma_X\sigma_Ysin\theta *cos\theta $


b) $ cov(Z,W) = E[Z,W] = [(X cos \theta + Y sin \theta)*(Y cos\theta - X sin\theta)] $

$ = E[Y^2]sin\theta cos\theta - E[X^2] sin\theta cos\theta + E[XY]*(cos^2\theta - sin^2\theta) $

$ = \frac{1}{2} sin2\theta *(\sigma_Y^2 - \sigma_X^2)+cos 2\theta* \rho\sigma_X\sigma_Y $

Alumni Liaison

ECE462 Survivor

Seraj Dosenbach