Mean Distances in p-dimensions

In my last post, I calculated the mean distance to the nearest point to the origin among \(N\) points taken from a uniform distribution on \([-1, 1]\). This turned out to be \(r = 1/(N+1)\), which is close to but greater than the median distance \(m = 1 - (1/2)^{1/N}\). In this post, I want to generalize the calculation of the mean to a \(p\)-dimensional uniform distribution over the unit ball.

(Mean) Distances in Uniform Distributions

Consider the uniform distribution on \([-1, 1]\), and take \(N\) points from this distribution. What is the mean distance from the origin to the nearest point? If you take the median instead of the mean, you get the answer outlined in my last post. The mean makes things more challenging. Here is a solution that makes sense to me. I am sure there is a more formalized way to go about this, but I was trained as a physicist, so I tend to use “informal” mathematics.

(Median) Distances in Uniform Distributions

Take a uniform distribution on \([-1, 1]\), so \(p(x) = 1/2\). Pick a single point from this distribution. The probability that this point is within a distance \(m\) from the origin is \(2m/2 = m\). The probability that this point is not within a distance \(m\) from the origin is then \(1-m\). Now consider picking \(N\) points from the distribution. The probability that all \(N\) points are further than \(m\) from the origin is \((1-m)^N\).

A Linear Model for Linda

What is Linda? Data Cleaning Linda Finish and Bodyweight (Men) Linda Finish and Bodyweight (Women) What is Linda? Linda is the name of a CrossFit benchmark workout. The original form of the workout is 10-9-8-7-6-5-4-3-2-1 reps of the triplet deadlift at 1.5 times bodyweight bench press at bodyweight clean at 0.75 times bodyweight Linda was included in the 2018 CrossFit regionals in a standardized form.