calculus and analysis - Solve this probability problem symbolically

Consider a unit square, Pick two points P and Q uniformly at random inside the square, What is the probability that |PQ|>1?

I tried solve this problem

Integrate[Boole[(x1-x2)^2+(y1-y2)^2>1],{x1,0,1},{y1,0,1},{x2,0,1},{y2,0,1}]

above code with NIntegrate given 0.025074 , but I want a symbolically result as except is $\frac{19}{6}-\pi$ .

Answer

This is not quick (includes J.M. comment):

pdf = UniformDistribution[2];
td = TransformedDistribution[(x - y)^2, {x, y} \[Distributed] pdf];
zd = TransformedDistribution[
   a + b, {a \[Distributed] td, b \[Distributed] td}];

then

ans = 1 - FullSimplify[CDF[zd,1]]

yields the desired result.

Comments

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