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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

$$ \frac{3\pi/4+1/2}\pi=\frac34+\frac1{2\pi}\;. $$

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Thanks!! That's exactly the answer but I am wondering how we can get there through integrating the density function of x and y? – Inia Oct 26 '12 at 16:30
@Inia: Well, that's essentially the question how to find the areas of a circle and a triangle by integration -- you can find a lot about that by Googling something like "circle area by integration", e.g. Wikipedia. – joriki Oct 26 '12 at 19:22

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