Suppose that x and y are zero-mean jointly gaussian random variables with variances and , respectively, and correlation coefficient . the joint pdf surface is a flattened bell. now, suppose we rotate the coordinate axes by an angle . the random point in the plane whose coordinates are (x, y) with respect to the old axes now has coordinates (z, w) with respect to the new axes where , and . the joint pdf surface is still the same shape as before, but since the coordinate axes have been changed, the pdf surface is defined by a different formula.
(a)find the means and variances of the random variables z and w, and compute . from these values, write the new formula for the joint pdf surface.
(b)for suitable choice(s) of , the new coordinate axes are parallel to the major and minor axes of the ellipses that are the contours of the joint pdf surface. for these choices of , z and ware independent gaussian random variables. find all angles such that z and w are independent gaussian