Consider the circle $\large\lambda$ of radius $1$ centered at $(1,1)$.
Say I drop a pin directly onto $(x,y) \in \large \lambda$.
What is the expected value of $xy$?
A part of me was like $1$ x $1 = \boxed1$.
Another part of me was like $\displaystyle \frac{1}{π} \int_0^2 \int_{1-\sqrt{1-(y-1)^2}}^{1+\sqrt{1-(y-1)^2}}xy\,dx\,dy=\boxed1$.
Another piece of me was like $\displaystyle \frac{1}{2π}\int_0^{2π}(\sin x+1)(\cos x+1)\,dx=\boxed1$ .
Here was the problem verbatim:
Compute the expected value of the product of the coordinates of a point randomly selected on a circle of radius $1$ centered at the point $(1,1)$.
So my ideas were
- Average the coordinates and multiply
- Find the average over the circle
- Find the average over the perimeter
And somehow all of them ended up with the same answer. Trippy.
Which one is correct? I felt that the problem was a bit vague.
A point on the circle is a point of the form $(1+\cos \theta, 1+\sin \theta)$, so your second integral is the correct formulation of the problem. As the trig functions average to zero over a period, the integral is $1$ as you say.