Fall 2011 Homework 8: Question 6
Let X, Y be continuous r.v.s with a spherically symmetric joint distribution, which means that the joint PDF is of the form $f(x, y)=g(x^{2}+y^{2})$ for some function g. Let (R, $\theta$) be the polar coordinates of (X, Y), so $R^{2}=X^{2}+Y^{2}$ is the squared distance from the origin and $\theta$ is the angle (in [0, 2$\pi$)), with X = Rcos $\theta$, Y = Rsin $\theta$.
(a) Explain intuitively why R and $\theta$ are independent. Then prove this by finding the joint PDF of (R, $\theta$).
(b) What is the joint PDF of (R, $\theta$) when (X, Y) is Uniform in the unit disc {(x, y) : $x^{2}+y^{2}\leq 1$}?
(c) What is the joint PDF of (R, $\theta$) 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.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."