Is it true, that if $U_{1}$ and $U_{2}$ are iid uniform distributed variables on $\left[-1,1\right]$, then the sum of $U_{1}^{2}$ and $U_{2}^{2}$ is still uniform distributed conditioned on the set that this sum is not greater than $1$? Other words:
$$X\dot{=}U_{1}^{2}+U_{2}^{2}|U_{1}^{2}+U_{2}^{2}\leq1\sim Uni\left[0,1\right]?$$
I've read this in a book and for first glance this hasn't been clear. I guess we have to compare some kind of “conditional charachteristic” functions to decide this question.
Best Answer
A geometric proof: after conditioning, $(U_1, U_2)$ is a point chosen uniformly at random from the unit disk.
So for $0 < x < 1$, the probability $P(U_1^2 + U_2^2 < x)$ is the area of a disk of radius $\sqrt{x}$ divided by the area of the unit disk.
That's $\pi (\sqrt{x})^2/\pi = x$, giving the result.