Distribution of $X^2+Y^2$ where $X,Y$ are independent $U[0,1]$ variables

Question

Let $X$ and $Y$ be independent random variables which are uniformly distributed over the interval $[0,1]$. Find the density of the random variable $Z = X^{2} + Y^{2}$.

Should I compute the density of $X^{2}$ and $Y^{2}$ first? And how to solve it?

Answer 1

For $1 \le z \le 2$, the region $x^2 + y^2 \le z$, $0 \le x,y \le 1$ looks like this:

The two right triangles have height $\sqrt{z-1}$ and base $1$, so total area $\sqrt{z-1}$. The sector has opening angle $\pi/2 - 2 \arctan(\sqrt{z-1})$ and radius $\sqrt{z}$, so area $(\pi/2 - 2 \arctan(\sqrt{z-1})) z/2$. Thus $$ \mathbb P(X^2 + Y^2 \le z) = \sqrt{z-1} + (\pi/2 - 2 \arctan(\sqrt{z-1})) z/2$$ The density of $X^2 + Y^2$ in this interval is the derivative of that, or $$ \pi/4 - \arctan(\sqrt{z-1})$$