Finding the covariance of coordinates of a circle.

Question

Let $C$ be a circle around the origin with radius $R$. Let $R = r$ be fixed. Take $U$ uniformly distributed on the circumference of $C$ and write $U$ = $(X, Y)$, where $X$ and $Y$ are the Cartesian coordinates of $U$. Find $\mathrm{Cov}(X, Y)$. Are $X$ and $Y$ independent?

Answer 1

You might be able to get a little bit more insight with a picture:

Shown is a circle with $r=2$ around the origin. By definition of $\sin\Theta$ and $\cos\Theta$ you can see that $(X_u,Y_u) = (r\cos\Theta,r\sin\Theta)$. So clearly $X$ and $Y$ are not independent, because $X=r\cos(\arcsin\left(\frac{Y}r\right))$.

You probably know $\mathrm{Cov}(X,Y) = \mathsf{E}[XY]-\mathsf{E}[X]\mathsf{E}[Y].$ Well, we can find all of these expectations because we now have formulas for each of them from $\Theta$, and we know $\Theta \sim \mathrm{Unif}(0,2\pi).$ I'll show you $\mathsf{E}[X]:$

$$\mathsf{E}[X] = \mathsf{E}[r\cos\Theta] = \int_0^{2\pi}r\cos(\theta)f_{\Theta}(\theta)d\theta = \int_0^{2\pi}r\cos(\theta)\cfrac{1}{2\pi}d\theta = 0$$

Hope that helps.

Answer 2

If you pick a value for X, you have just 2 options for Y that would make that point to land on the circle. In this case X and Y are not independent.