## The expected area of a triangle

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serur
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### The expected area of a triangle

Given a unit circle, we pick independently 3 points A,B, and C (uniform distribution) on its circumference. What is the expected area of a triangle ABC?
What I know so far: density - 1/(2pi), picking a point is just the same as picking an angle uniformly distributed over [-pi,pi], and the area is something like 8|sin(i-j)*sin(j-k)*sin(k-i)|...

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serur
A great helper
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Joined: Thu Feb 23, 2006 11:30 pm
EDIT: you're not likely to be drawn into discussion of a kindergarten problem I see I have this question about densities:
Given random variables X, Y, (oo >X,Y >=0) and their joint probability density function f(x,y); consider a random variable Z := X*Y. What's the probability density of Z? The answer is "Integral from 0 to oo over (1/x)f(x,z/x)dx". Now, the intuition behind densities says that
f(x,y)dxdy ~ probability of (x,y) falling into the rectangle [x,x+dx]x[y,y+dy]. What's the meaning of a (1/x) ?