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علماء الرياضيات | Uniform Sum Distribution |
 2007
 02:50 صباحاً
date: 16-4-2021 |
Author : Derbyshire, J. 
Book or Source : Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004. 
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The distribution for the sum 
of 
uniform variates on the interval ![[0,1]](https://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline3.gif)
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),](https://mathworld.wolfram.com/images/equations/UniformSumDistribution/NumberedEquation2.gif) |
(2) |
where the Fourier parameters are taken as 
. The first few values of 
are then given by
 |
 |
![1/2[sgn(1-u)+sgnu]](https://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline9.gif) |
(3) |
 |
 |
![1/2[(-2+u)sgn(-2+u)-2(-1+u)sgn(-1+u)+usgnu]](https://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline12.gif) |
(4) |
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![1/4[-(-3+u)^2sgn(-3+u)+3(-2+u)^2sgn(-2+u)-3(-1+u)^2sgn(-1+u)+u^2sgnu]](https://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline15.gif) |
(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],](https://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline18.gif) |
(6) |
illustrated above.
Interestingly, the expected number of picks 
of a number 
from a uniform distribution on ![[0,1]](https://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline21.gif)
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)!)]](https://mathworld.wolfram.com/images/equations/UniformSumDistribution/Inline30.gif) |
(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) |
 |
 |
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(14) |
 |
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(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.
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