Expected Distance Between Sphere Points

First, by the cosine law we note that the distance $d$ from $(0,0,1)$ to any point on the unit-sphere with polar angle $\theta$ is given by

$d^{2} = 1^{2} + 1^{2} - 2 \times 1 \times 1 \times \cos \theta = 2(1 - \cos \theta) = 2 \times 2\sin^{2} \dfrac{\theta}{2} \Longrightarrow d = 2\sin \dfrac{\theta}{2}$ ,

where the identity $\cos(2x) = 1 - 2\sin^{2}(x)$ was used. So by integration using spherical coordinates with $r = 1$ , we obtain a desired expected distance of

$\displaystyle \dfrac{1}{4\pi} \int_{\phi = 0}^{2\pi} \int_{\theta = 0}^{\pi} 2\sin \dfrac{\theta}{2} \space \sin \theta \space d\theta \space d\phi = \int_{0}^{\pi} 2 \sin^{2} \dfrac{\theta}{2} \cos \dfrac{\theta}{2} \space d\theta = 4 \int_{0}^{1} u^{2} du = \dfrac{4}{3} = \boxed{1.333....}$ ,

where the identity $\sin \theta = 2 \sin \dfrac{\theta}{2} \space \cos \dfrac{\theta}{2}$ and the substitution $u = \sin \dfrac{\theta}{2}$ were used.

Note: The averaging factor of $4 \pi$ represents the surface area of a unit sphere.