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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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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