Uniform Sum Distribution

The distribution for the sum  of  uniform variates on the interval  can be found directly as

(1)

where  is a delta function.

A more elegant approach uses the characteristic function to obtain

(2)

where the Fourier parameters are taken as . The first few values of  are then given by

			(3)
			(4)
			(5)
			(6)

illustrated above.

Interestingly, the expected number of picks  of a number  from a uniform distribution on  so that the sum  exceeds 1 is e (Derbyshire 2004, pp. 366-367). This can be demonstrated by noting that the probability of the sum of  variates being greater than 1 while the sum of  variates being less than 1 is

			(7)
			(8)
			(9)

The values for , 2, ... are 0, 1/2, 1/3, 1/8, 1/30, 1/144, 1/840, 1/5760, 1/45360, ... (OEIS A001048). The expected number of picks needed to first exceed 1 is then simply

(10)

It is more complicated to compute the expected number of picks that is needed for their sum to first exceed 2. In this case,

			(11)
			(12)

The first few terms are therefore 0, 0, 1/6, 1/3, 11/40, 13/90, 19/336, 1/56, 247/51840, 251/226800, ... (OEIS A090137 and A090138). The expected number of picks needed to first exceed 2 is then simply

			(13)
			(14)
			(15)

The following table summarizes the expected number of picks  for the sum to first exceed an integer  (OEIS A089087). A closed form is given by

(16)

(Uspensky 1937, p. 278).

		OEIS	approximate
1		A001113	2.71828182...
2		A090142	4.67077427...
3		A090143	6.66656563...
4		A089139	8.66660449...
5		A090611	10.66666206...

REFERENCES:

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.

Sloane, N. J. A. Sequences A001048/M0890, A001113/M1727, A089087, A089139, A090137, A090138, A090142, A090143, and A090611 in "The On-Line Encyclopedia of Integer Sequences."

Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, 1937.