👁 Image

Uniform Sum Distribution

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👁 UniformSumDistribution

The distribution for the sum 👁 X_1+X_2+...+X_n
of 👁 n
uniform variates on the interval 👁 [0,1]
can be found directly as

👁 P_(X_1+...+X_n)(u)=intint...int_()_(n)delta(x_1+x_2+...+x_n-u)dx_1dx_2...dx_n,

(1)

where 👁 delta(x)
is a delta function.

A more elegant approach uses the characteristic function to obtain

👁 P_(X_1+...+X_n)(u)=F_t^(-1)[((i(1-e^(it)))/t)^n](u) =1/(2(n-1)!)sum_(k=0)^n(-1)^k(n; k)(u-k)^(n-1)sgn(u-k),

(2)

where the Fourier parameters are taken as 👁 (1,1)
. The first few values of 👁 P_n(u)
are then given by

👁 P_(X_1)(u)	👁 =	👁 1/2[sgn(1-u)+sgnu]	(3)
👁 P_(X_1+X_2)(u)	👁 =	👁 1/2[(-2+u)sgn(-2+u)-2(-1+u)sgn(-1+u)+usgnu]	(4)
👁 P_(X_1+X_2+X_3)(u)	👁 =	👁 1/4[-(-3+u)^2sgn(-3+u)+3(-2+u)^2sgn(-2+u)-3(-1+u)^2sgn(-1+u)+u^2sgnu]	(5)
👁 P_(X_1+X_2+X_3+X_4)(u)	👁 =	👁 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],	(6)

illustrated above.

Interestingly, the expected number of picks 👁 n
of a number 👁 x_k
from a uniform distribution on 👁 [0,1]
so that the sum 👁 sum_(k=1)^(n)x_k
exceeds 1 is e (Derbyshire 2004, pp. 366-367). This can be demonstrated by noting that the probability of the sum of 👁 n
variates being greater than 1 while the sum of 👁 n-1
variates being less than 1 is

👁 P_n^((1))	👁 =	👁 int_1^nP_(X_1+...+X_n)(u)du-int_1^(n-1)P_(X_1+...+X_(n-1))(u)du	(7)
👁 Image	👁 =	👁 (1-1/(n!))-[1-1/((n-1)!)]	(8)
👁 Image	👁 =	👁 1/(n(n-2)!).	(9)

The values for 👁 n=1
, 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

👁 <n_1>=sum_(n=1)^inftynP_n^((1))=sum_(n=1)^infty1/((n-2)!)=sum_(n=0)^infty1/(n!)=e.

(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,

👁 P_n^((2))	👁 =	👁 int_2^nP_(X_1+...+X_n)(u)du-int_2^(n-1)P_(X_1+...+X_(n-1))(u)du	(11)
👁 Image	👁 =	👁 ((n-2)(2^(n-1)-n))/(n!).	(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

👁 <n_2>	👁 =	👁 sum_(n=1)^(infty)nP_n^((2))	(13)
👁 Image	👁 =	👁 sum_(n=1)^(infty)(n(n-2)(2^(n-1)-n))/(n!)	(14)
👁 Image	👁 =	👁 e^2-e.	(15)

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

👁 <n_s>=1/(n!)sum_(k=0)^n((-1)^kn!(n-k+1)^k)/(k!)e^(n-k+1)

(16)

(Uspensky 1937, p. 278).

👁 s	👁 <n_s>	OEIS	approximate
1	👁 e	A001113	2.71828182...
2	👁 e^2-e	A090142	4.67077427...
3	👁 1/2(2e^3-4e^2+e)	A090143	6.66656563...
4	👁 1/6(6e^4-18e^3+12e^2-e)	A089139	8.66660449...
5	👁 1/(24)(24e^5-96e^4+108e^3-32e^2+e)	A090611	10.66666206...

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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.

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Uniform Sum Distribution

Cite this as:

Weisstein, Eric W. "Uniform Sum Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UniformSumDistribution.html

URL: https://mathworld.wolfram.com/UniformSumDistribution.html