1 Let X and Y be independent standard normal random variable

(1) Let X and Y be independent standard normal random variables. Define

Xˆ=X / (X2 +Y2)

Yˆ=Y / (X2 +Y2)

Use Xˆ = R cos and Yˆ = R sin for R 0, 0 < 2 and find the joint cdf for

R and . (Hint: R and are not jointly continuous).

joint pdf of X` AND Y` = e ^ - [ [ X2 / 2(X2+Y2) ] + [ Y2 / 2 (X2+Y2) ] ] / 2*pi..

X` =cos and Y` = sin..

Jacobian = R...

joint pdf = e^(-1/2)* R / 2...when R 0, 0 < 2
= 0 OTHERWISE....