Independent Gaussian Question - showing ISN

by Alex Thornhill   Last Updated May 23, 2020 01:19 AM

Suppose that X and Y are independent random variables each with the standard normal distribution. Let X = RcosΘ and Y = RsinΘ be the polar coordinate representation of the point (X,Y ), with the angular coordinate Θ chosen so that 0 ≤ Θ < 2π.

I found density of Y/X by assigning Z=Y/X and using the fact that the quotient of two normal RVs = cauchy distribution.

But now I'm asked to Show that X = Rcos2Θ and Y = Rsin2Θ are independent standard normal random variables, and to also use that to show that the random variables 2XY/(√X^2 + Y^2) and (X^2 −Y^2)/(√X^2 + Y^2) are independent standard normal random variables. Not sure how to go about this

Tags : probability

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