Vtyjkyuk

Suppose we've picked two points randomly from a uniform distribution over the Euclidean plane and we know that the Euclidean distance between them is $d$. What is the expected value of the Manhattan distance, $m$, between the two points, $E[m|d]$?

For context: I originally thought it was just a simple application of the Pythagorean theorem. But knowing that $a^2+b^2=c^2$ doesn't allow me to recover $a+b$ as a function of $c$, right? That function depends on factors other than the side lengths, and it's not immediately obvious to me as to how to proceed. For further context, I have haversine distances between pairs of points. I'd love a quick and dirty way to convert these to driving distances, for which Manhattan would work fine assuming a grid-like transportation network.

Basic approach. Make the assumption that one point is at the origin, and the other point is uniformly distributed on the circle centered at the origin with radius $d$. Equivalently, the argument $theta$ of that point is uniformly distributed on the interval $[0, 2pi]$. Note that the Manhattan distance $m$ is $|dsintheta| + |dcostheta|$, so

$$
E(m mid d) = E(|dsintheta|)+E(|dcostheta|)
$$

Simple integration should take you the rest of the way there—does that suffice, or could you use more direction?

What you are asking is simply the average Manhattan distance of the points on a circle of radius $d$ to the center. By symmetry you can perform the computation in a single quadrant and evaluate the average abscissa and average ordinate, which are equal.

In polar coordinates and for a unit circle,

$$fracpi2 E(x)=int_0^pi/2 x,dtheta=int_0^pi/2 costheta,dtheta=1.$$

Finally,

$$E(m)=frac 4pi d$$ where the factor is larger than one and smaller than the square root of two, as could be expected.