Let X, Y be continuous r.v.s with a spherically symmetric joint distribution, which means that the joint PDF is of the form for some function g. Let (R, ) be the polar coordinates of (X, Y), so is the squared distance from the origin and is the angle (in [0, 2)), with X = Rcos , Y = Rsin .

(a) Explain intuitively why R and are independent. Then prove this by finding the joint PDF of (R, ).

(b) What is the joint PDF of (R, ) when (X, Y) is Uniform in the unit disc {(x, y) : }?

(c) What is the joint PDF of (R, ) when X and Y are i.i.d. N(0, 1)?

(a) Explain intuitively why R and are independent. Then prove this by finding the joint PDF of (R, ).

(b) What is the joint PDF of (R, ) when (X, Y) is Uniform in the unit disc {(x, y) : }?

(c) What is the joint PDF of (R, ) when X and Y are i.i.d. N(0, 1)?

Solution:
(a) Intuitively, this makes sense since the joint PDF of X, Y at a point (x, y) only depends on the distance from (x, y) to the origin, not on the angle, so knowing R gives no information about theta. The joint factors as a function of r times a (constant) function of t, so R and theta are independent with theta ~ Unif(0, 2pi). (b) r/pi, where R and theta are independent. (c) (1/2pi)re^(−r^2/2) where R and theta are independent and the distribution of R is called Weibull.

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