Radial Joint Probability
Suppose two random variables X and Y are uniformly distributed over the disk X 2 + Y 2 ≤ 4 . Find the probability that a particular sample of ( X , Y ) lies in the annulus of radius greater than 1 and less than 2.
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1 solution
Can you do this using joint integrals like in the example above? When I tried with x and y between -2 and 2 and then between -1 and 1, I didn't get a fraction that would give 3/4. Instead I got 8/128.
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Sure, you can do it with multiple integrals, but you would need to be careful to do it in polar coordinates and to understand that the uniform distribution looks like 1/Area ( 4 π 1 in this case). The problem with your approach is that the region your integrals describe is not an annulus, but rather a square with a square-shaped hole in it. The correct integrals in this case would be: P = ∫ 0 2 π d θ ∫ 1 2 4 π 1 r d r = 4 π 1 2 ( 4 − 1 ) ( 2 π ) = 4 3
Since X and Y are uniformly distributed, the probability of any point being in a particular region is the area of that region over the total area. The area of the region with radius greater than 1 and less than 2 is 4 π − π = 3 π . The total area of the region from which X and Y are drawn is 4 π . Thus the probability of finding ( X , Y ) in this annulus is 4 π 3 π = 4 3 .