(New page: E[X] = E[Y] = 0 <math>E[X^2] = _sigma^2) |
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| − | E[X] = E[Y] = 0 | + | <math>E[X] = E[Y] = 0</math> |
| − | <math>E[X^2] = | + | |
| + | <math>E[X] = \sigma_X^2</math> | ||
| + | |||
| + | <math>E[y] = \sigma_Y^2</math> | ||
| + | |||
| + | cov[X,Y] = <math>\rho * \sigma_X * \sigma_Y</math> | ||
| + | |||
| + | <math>E[Z] = E[X cos \theta + Y sin \theta] = E[X]cos\theta + E[Y]sin\theta = 0</math> | ||
| + | |||
| + | <math>E[W] = E[Ycos\theta - X sin\theta] = 0</math> | ||
| + | |||
| + | <math>Var(Z) = E[Z^2] = E[X^2 cos^2\theta + Y^2 sin^2\theta + 2*\rho\sigma_x\sigma_y]</math> | ||
| + | |||
| + | <math>Var(Z) = \sigma_X^2 cos^2\theta+\sigma_Y^2sin^2\theta+2*\rho\sigma_X\sigma_Ysin\theta *cos\theta</math> | ||
| + | |||
| + | <math>Var(W) = \sigma_X^2 sin^2\theta+\sigma_Y^2cos^2\theta-2*\rho\sigma_X\sigma_Ysin\theta *cos\theta</math> | ||
| + | |||
| + | |||
| + | b) <math>cov(Z,W) = E[Z,W] = [(X cos \theta + Y sin \theta)*(Y cos\theta - X sin\theta)]</math> | ||
| + | |||
| + | <math> = E[Y^2]sin\theta cos\theta - E[X^2] sin\theta cos\theta + E[XY]*(cos^2\theta - sin^2\theta)</math> | ||
| + | |||
| + | <math> = \frac{1}{2} sin2\theta *(\sigma_Y^2 - \sigma_X^2)+cos 2\theta* \rho\sigma_X\sigma_Y</math> | ||
Latest revision as of 16:41, 9 December 2008
$ 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 $
