Expected Distance on a Circle

Calculus Level 5

An angle \(\theta\) is chosen randomly in the interval \([0,2\pi )\), and its corresponding point on the unit circle is plotted. Let this point be \(P\). Now let \(A=(1,0)\) and \(B=(-1,0)\).

Let the expected value of \(PA+PB-2\) be \(E\).

What is \(\lfloor 1000E\rfloor\)?

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