What is the probability that the range of a set of N randomly chosen real numbers in [0, 1] is less than the reciprocal of N?

(Random number with uniform distribution over [0, 1])

For clarification, in the case where N = 2, we can use geometric probability. On the coordinate plane consider points with 0<=x,y<=1. The condition is satisfied on a diagonal band of area 3/4 from the origin to (1,1). Similarly, with N = 3 the volume of the space in which the condition is satisfied is 7/27.

• How did you compute 7/27, and why did your argument not generalize to arbitrary N? Jan 5 '11 at 21:23
• What is meant by "the range of a set"? Is that just max minus min? Jan 6 '11 at 4:03

The keyword is the order statistics. The distributions of the maximum and minimum values of a sample of $n$ independent uniformly distributed random variables are given respectively by the laws
$$U_{max}\sim \mbox{Beta}(n,1),\qquad U_{min}\sim \mbox{Beta}(1,n).$$
The range $U_{max}-U_{min}$ has a $\mbox{Beta}(n-1,2)$ distribution (see, e.g., Section 2.5 of A First Course in Order Statistics) so $$\mathbb P\{U_{max}-U_{min} < a\}=\frac{1}{B(n-1,2)}\int_{0}^{a}x^{n-2}(1-x)dx=na^{n-1}-(n-1)a^n.$$
• =$(n^2-n+1)n^{-n}$ which agrees with the examples given (however arrived at) Jan 5 '11 at 22:23
• (Could I suggest writing $U_\max$ and $U_\min$ instead of $U_{max}$ and $U_{min}$? The TeX code is U_\max and U_\min. Using \max and \min not only prevents italicization, but also causes standard formatting conventions to be followed in some contexts.) Jan 6 '11 at 18:40