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. 366367). 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).