There are 2 Random samples from [0, 1] uniform distribution. What is the average distance between them.

If we call u1, u2 our uniform randoms, then the distance is z:=|u2-u1| and is constrained to 0<z<1. The distribution function is:The expectation easily follows:

pterodactyl, a hint: actually all three segments have the average length of 1/3.here is a clever symmetry argument due to roah: consider instead picking three random points on a circle. by symmetry, each segment has an average length = 1/3 of the total circumference. note only the relative positions matter on a circle - one of the points could be picked to cut the loop open (symmetry broken) hence the desired results

MS5 -- how did you come up with the distribution fn 2(1-z). Cut point logic is too good.

QuoteOriginally posted by: pterodactylMS5 -- how did you come up with the distribution fn 2(1-z). Cut point logic is too good.Compute the cumulative distribution function and take its derivative. If you have the unit square in coordinates (x,y), then one has,where the factor of 2 comes from symmetry.

I haven't figured out the other solutions yet, but the way I did it was:consider the first point x between 0 and 1, then when the second point is greater than x, the average distance between them will be (1-x)/2. And when the second point is less than x, the average distance between them will be x/2. Then the average distance across both sides will be the weighted average of the two terms. The weights will be (1-x) and x. Then integrate the weighted average over 0->1

QuoteOriginally posted by: wileyswpterodactyl, a hint: actually all three segments have the average length of 1/3.here is a clever symmetry argument due to roah: consider instead picking three random points on a circle. by symmetry, each segment has an average length = 1/3 of the total circumference. note only the relative positions matter on a circle - one of the points could be picked to cut the loop open (symmetry broken) hence the desired resultsThat's one of those solutions where everbody goes, "ohh"So if it were three random points on a line, the average distance is 1/4?

kwd1, yes, if these three points are independent and uniformly distributed