average of sign function

Question

Suppose we are given a unit vector $\vec{p}$, and a unit vector $\vec{\lambda}$ uniformly distributed on the hemisphere $\vec{p} \cdot \vec{\lambda} >0$. Further, let $\vec{a'}$ be a vector whose angle with $\vec{p}$ is $\theta'$. I am asked to show that the average of $\text{sign}(\vec{\lambda}\cdot \vec{a')}$ over $\vec{\lambda}$ is $1-2 \theta'/\pi$. How can I go about this?

Answer 1

Consider the plane $\Pi$ through the origin perpendicular to $\vec{a}$. Then all vectors $\vec{\lambda}$ on the same side of $\Pi$ as $\vec{a}$ satisfy $\vec{\lambda} \cdot \vec{a} >0$ since the angle between them is less than $\frac{\pi}{2}$. Vectors on the other side satisfy $\vec{\lambda} \cdot \vec{a} <0$. So we only need to find the areas of the sections that $\Pi$ splits the hemisphere into. But the section on the same side as $\Pi$ contains $1-\frac{\theta ’}{\pi}$ of the surface, and the other section contains $\frac{\theta ’}{\pi}$ of the surface, so the average is $1-\frac{\theta ’}{\pi}-\frac{\theta ’}{\pi} = 1-\frac{2 \theta ’}{\pi}$, as required.