The distribution for the sum 
of 
uniform variates on the interval ![[0,1]](http://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline3.svg)
can be found directly as
 | (1) |
where 
is a delta function.
A more elegant approach uses the characteristic function to obtain
 =1/(2(n-1)!)sum_(k=0)^n(-1)^k(n; k)(u-k)^(n-1)sgn(u-k),](http://mathworld.wolfram.com/images/equations/UniformSumDistribution/NumberedEquation2.svg) | (2) |
where the Fourier parameters are taken as 
. The first few values of 
are then given by
 |  | ![1/2[sgn(1-u)+sgnu]](http://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline9.svg) | (3) |
 |  | ![1/2[(-2+u)sgn(-2+u)-2(-1+u)sgn(-1+u)+usgnu]](http://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline12.svg) | (4) |
 |  | ![1/4[-(-3+u)^2sgn(-3+u)+3(-2+u)^2sgn(-2+u)-3(-1+u)^2sgn(-1+u)+u^2sgnu]](http://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline15.svg) | (5) |
 |  | ![1/(12)[(-4+u)^3sgn(-4+u)-4(-3+u)^3sgn(-3+u)+6(-2+u)^3sgn(-2+u)-4(-1+u)^3sgn(-1+u)+u^3sgnu],](http://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline18.svg) | (6) |
illustrated above.
Interestingly, the expected number of picks 
of a number 
from a uniform distribution on ![[0,1]](http://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline21.svg)
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) |
 |  | ![(1-1/(n!))-[1-1/((n-1)!)]](http://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline30.svg) | (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).
