If $X,Y$ ~$U(0,1)$ what is the distribution of $Z=0.5x^{2}+0.5y^{2}$?

Question

I have some trouble with it.. the question is: $X,Y$ uniformly distributed $U(0,1)$ than $\frac{1}{2}(x^2+y^2) $~$exp(1)$... I am not even sure it is correct.. I know that if $X,Y$~$N(0,1)$ than it is correct.

Answer 1

$exp(1)$ is clearly incorrect. Since $X$, $Y$ are bounded then $\frac{1}{2}(X^2+Y^2)$ is also bounded, so it can't be exponential.

Answer 2

Think of the gap between circles of radius $r$ and $r+dr$, and where they intersect the unit square.

The distribution is $P(A)=Pr(A<(x^2+y^2)/2<A+dA)/dA=Pr(\sqrt{2A}<r<\sqrt{2A}+dA/\sqrt{2A})/dA$

If $r<1$, then the area of this gap is for a quarter-circle, area $(\pi/2)rdr=(\pi/2)\sqrt{2A}dA/\sqrt{2A}$, so the distribution is $P(A)=\pi /2$ for $0<A<1/2$.

If $1<r<\sqrt{2}$, then the angle is trimmed to $\pi/2-2\arccos (1/r)$, so the area is that angle times $rdr$, and the distribution is $(\pi/2-2\arccos(1/\sqrt{2A}))$ for $1/2<A<1$.