Question

3. A point (X, Y) is uniformly distributed on the unit square (0, 1]2. Let 0 be the angle between the r-axis and the line segment that connects (0,0) to the point (X, Y). Find the expected value El9] (Hint: recall that conin 0 and an

Answer 1

$tan (\theta) = \frac{y}{x}$

$\int _0^1\int _0^1\arctan \left(\frac{y}{x}\right)dxdy=\frac{\pi }{4}\quad \left(\mathrm{Decimal:\quad }\:0.78540\dots \right)$

$\int _0^1\arctan \left(\frac{y}{x}\right)dx=\frac{y\ln \left(y^2+1\right)-y\ln \left(y^2\right)}{2}+\arctan \left(y\right)$

$\int _0^1\left(\frac{y\ln \left(y^2+1\right)-y\ln \left(y^2\right)}{2}+\arctan \left(y\right)\right)dy=\frac{\pi }{4}$