How do you uniformly sample spans from a bounded line?

Question

Suppose you have a bounded and continuous line. For example, the line could include all real numbers between 0 and 3. How do you sample spans from the line such that...

• Any point on the line has an equal chance of being included in the span.
• The distribution of span lengths is uniform.

---

My attempted solution:

import random

max_length = 15.0
bounds = (0.0, 3.0)

length = random.uniform(0, max_length)
start = random.uniform(bounds[0] - length, bounds[1])
end = min(start + length, bounds[1])
start = max(start, bounds[0])

This solution is able to fulfill the first criterion and not the second criterion.

EDIT: Following the lead of BruceET, I have plotted the distribution of span lengths:

import random
from matplotlib import pyplot
import seaborn

max_length = 6
bounds = (0, 3)
num_samples = 10**6

samples = []
for _ in range(num_samples):
    length = random.uniform(0, max_length)
    start = random.uniform(bounds[0] - length, bounds[1])
    end = min(start + length, bounds[1])
    start = max(start, bounds[0])
    samples.append(end - start)

seaborn.distplot(
    samples,
    hist=True,
    kde=True,
    bins=30,
    color='darkblue',
    hist_kws={'edgecolor': 'black'},
    kde_kws={'linewidth': 1})
pyplot.show()

Answer 1

Comment: You're having trouble with your second simulation because the 'span' of several uniform random variables is is not uniform.

In R, the function range gives the endpoints, taking the difference gives what you call the 'span'.

Here is a simulation in R of the span lengths from 100,000 samples of size $n=5$ from $\mathsf{Unif}(0, 3).$

set.seed(2020)
span.5 = replicate(10^6, diff(range(runif(5,0,3))))
summary(span.5)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.07294 1.63716 2.05782 1.99968 2.41826 2.99932 
hist(span.5, prob=T, col="skyblue2")
curve((1/3)*dbeta(x/3, 4, 2), add=T, col="red", lwd=2)