# Probability of sum of two independent variables given joint density

Let $x$ and $y$ be $2$ independent random vectors on the unit disk such that their joint density is just $\frac{1}{\pi}$. What is the probability that $x+y$ is less than $1$?

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Are you sure they are not independent? –  jay-sun Oct 26 '12 at 0:41
I suspect you mean that $x$ and $y$ are the coordinates of a point in the unit disk? However, in that case they're not independent. If you do mean vectors, how do you compare $x+y$ to $1$? –  joriki Oct 26 '12 at 0:43
Assuming that you mean that $x$ and $y$ are the coordinates of a point randomly uniformly chosen in the unit disk:
The area of the unit disk below the line $x+y=1$ consists of three quarter-circles with area $\pi/4$ each and a triangle with area $1/2$, so the probability is
$$\frac{3\pi/4+1/2}\pi=\frac34+\frac1{2\pi}\;.$$