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 $1$?
Probability of sum of two independent variables given joint density
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probability
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0Are you sure they are not independent? – 2012-10-26
1 Answers
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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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0@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](http://en.wikipedia.org/wiki/Area_of_a_disk). – 2012-10-26