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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)
=(1-1/(n!))-[1-1/((n-1)!)]
(8)
=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)
=((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)
=sum_(n=1)^(infty)(n(n-2)(2^(n-1)-n))/(n!)
(14)
=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>OEISapproximate
1eA0011132.71828182...
2e^2-eA0901424.67077427...
31/2(2e^3-4e^2+e)A0901436.66656563...
41/6(6e^4-18e^3+12e^2-e)A0891398.66660449...
51/(24)(24e^5-96e^4+108e^3-32e^2+e)A09061110.66666206...

See also

Uniform Difference Distribution, Uniform Distribution, Uniform Product Distribution, Uniform Ratio Distribution

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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 Web Resource. https://mathworld.wolfram.com/UniformSumDistribution.html

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