Distribution of functions of uniform random variables

Question

Given these two independent and uniform distributed random variables, $$X \sim U[-\pi,\pi]$$ and $$Y \sim U[-\pi,\pi]$$

What is the distribution of $$\sin(X)$$ and $$\sin (Y)$$ and the distribution of $$\sin(X)-\sin(Y)$$

Thanks

Answer 1

The number $\sin X$ is the coordinate on the vertical axis of the point $(\cos X,\sin X)$, which is uniformly distributed on the circle of unit radius centered at $(0,0)$. Since the distribution of the coordinate on the vertical axis on the left and right halves of the circle are the same, we may as well consider only the right half, so we have the distribution of $\sin X$ with $X$ uniformly distributed between $-\pi/2$ and $\pi/2$. This makes the sine function strictly increasing, so we can say $$ \Pr(\sin X\le w) = \Pr(X\le \arcsin w) = \frac{(\arcsin w) - (-\pi/2)}{(\pi/2)-(-\pi/2)} = \frac{(\arcsin w)+\pi/2}\pi. $$ The distribution of $Y$ is the same. And so is the distribution of $-Y$, since this distribution is symmetric about $0$. Hence the distribution of $\sin X-\sin Y$ is the same as that of $\sin X+\sin Y$. The density for that distribution can be found by convolving the density above with itself. Maybe I'll be back$\ldots\ldots$