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{{Probability distribution
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  |name      = Irwin-Hall distribution
  |type      = density
  |pdf_image  = [[Image:irwin-hall-pdf.svg|276px|Probability mass function for the distribution]]
  |cdf_image  = [[Image:irwin-hall-cdf.svg|276px|Cumulative distribution function for the distribution]]
  |parameters = ''n'' ∈ [[Natural numbers|'''N'''<sub>0</sub>]]
  |support    = <math>x \in [0,n]</math>
  |pdf        = <math>\frac{1}{(n-1)!}\sum_{k=0}^{\lfloor x\rfloor}(-1)^k\binom{n}{k}(x-k)^{n-1}</math>
  |cdf        = <math>\frac{1}{n!}\sum_{k=0}^{\lfloor x\rfloor}(-1)^k\binom{n}{k}(x-k)^n</math>
  |mean      = <math>\frac{n}{2}</math>
  |median    = <math>\frac{n}{2}</math>
  |mode      = <math>\begin{cases}
                  \text{any value in } [0,1]  & \text{for } n=1  \\
                  \frac{n}{2}              & \text{otherwise}
                \end{cases}</math>
  |variance  = <math>\frac{n}{12}</math>
  |skewness  = 0
  |kurtosis  = <math>-\tfrac{6}{5n}</math>
  |entropy    =
  |mgf        = <math>{\left(\frac{\mathrm{e}^{t}-1}{t}\right)}^n</math>
  |char      = <math>{\left(\frac{\mathrm{e}^{it}-1}{it}\right)}^n</math>
}}


In [[probability]] and [[statistics]], the '''Irwin–Hall distribution''', named after [[Joseph Oscar Irwin]] and [[Philip Hall]], is [[probability distribution]] for a [[random variable]] defined as sum of a number of [[statistically independent|independent]] random variables, each having a [[Uniform distribution (continuous)|uniform distribution]].<ref>Johnson, N.L.; Kotz, S.; Balakrishnan, N. (1995) ''Continuous Univariate Distributions'', Volume 2, 2nd Edition, Wiley ISBN 0-471-58494-0(Section 26.9)</ref> For this reason it is also known as the '''uniform sum distribution'''.
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The generation of [[Pseudo-random number sampling|pseudo-random number]]s having an approximately [[normal distribution]] is sometimes accomplished by computing the sum of a number of pseudo-random numbers having a uniform distribution; usually for the sake of simplicity of programming. Rescaling the Irwin–Hall distribution provides the exact distribution of the random variates being generated.
 
This distribution is sometimes confused with the [[Bates distribution]], which is the '''mean''' (not '''sum''') of n independent random variables uniformly distributed from 0 to&nbsp;1.
 
==Definition==
The Irwin–Hall distribution is the continuous [[probability distribution]] for the sum of ''n'' [[independent and identically distributed]] [[Uniform distribution (continuous)|''U''(0,&nbsp;1)]] random variables:
 
:<math>
X = \sum_{k=1}^n U_k.
</math>
 
The [[probability density function]] (pdf) is given by
 
:<math>
f_X(x;n)=\frac{1}{2\left(n-1\right)!}\sum_{k=0}^{n}\left(-1\right)^k{n \choose k}\left(x-k\right)^{n-1}\sgn(x-k)
</math>
 
where sgn(''x &minus; k'') denotes the [[sign function]]:
 
:<math> \sgn\left(x-k\right) = \begin{cases}
-1 &  x < k \\
0 &  x = k \\
1 &  x > k. \end{cases}
</math>
 
Thus the pdf is a [[spline (mathematics)|spline]] (piecewise polynomial function) of degree ''n''&nbsp;&minus;&nbsp;1 over the knots 0, 1, ..., ''n''. In fact, for ''x'' between the knots located at ''k'' and ''k'' + 1, the pdf is equal to
 
:<math>
f_X(x;n) = \frac{1}{\left(n-1\right)!}\sum_{j=0}^{n-1} a_j(k,n) x^j
</math>
 
where the coefficients ''a<sub>j</sub>(k,n)'' may be found from a [[recurrence relation]] over ''k''
 
:<math>
a_j(k,n)=\begin{cases} 1&k=0, j=n-1\\
                      0&k=0, j<n-1\\
a_j(k-1,n) + \left(-1\right)^{n+k-j-1}{n\choose
  k}{{n-1}\choose j}k^{n-j-1} &k>0\end{cases}
</math>
 
The coefficients are also [https://oeis.org/A188816 A188816] in [[On-Line Encyclopedia of Integer Sequences|OEIS]]. The coefficients for the cumulative distribution is [https://oeis.org/A188668 A188668].
 
The [[mean]] and [[variance]] are ''n''/2 and ''n''/12, respectively.
 
==Special cases==
* For ''n'' = 1, ''X'' follows a [[uniform distribution (continuous)|uniform distribution]]:
:<math>
f_X(x)= \begin{cases}
1        & 0\le x \le 1 \\
0      & \text{otherwise}
\end{cases}
</math>
* For ''n'' = 2, ''X'' follows a [[triangular distribution]]:
:<math>
f_X(x)= \begin{cases}
x        & 0\le x \le 1\\
2-x      & 1\le x \le 2
\end{cases}
</math>
* For ''n'' = 3,
:<math>
f_X(x)= \begin{cases}
\frac{1}{2}x^2                        & 0\le x \le 1\\
\frac{1}{2}\left(-2x^2 + 6x - 3 \right)& 1\le x \le 2\\
\frac{1}{2}\left(x^2 - 6x +9 \right)  & 2\le x \le 3
\end{cases}
</math>
* For ''n'' = 4,
:<math>
f_X(x)= \begin{cases}
\frac{1}{6}x^3                        & 0\le x \le 1\\
\frac{1}{6}\left(-3x^3 + 12x^2 - 12x+4 \right)& 1\le x \le 2\\
\frac{1}{6}\left(3x^3 - 24x^2 +60x-44 \right)  & 2\le x \le 3\\
\frac{1}{6}\left(-x^3 + 12x^2 -48x+64 \right)  & 3\le x \le 4
\end{cases}
</math>
* For ''n'' = 5,
:<math>
f_X(x)= \begin{cases}
\frac{1}{24}x^4                        & 0\le x \le 1\\
\frac{1}{24}\left(-4x^4 + 20x^3 - 30x^2+20x-5 \right)& 1\le x \le 2\\
\frac{1}{24}\left(6x^4-60x^3+210x^2-300x+155 \right)  & 2\le x \le 3\\
\frac{1}{24}\left(-4x^4+60x^3-330x^2+780x-655 \right)  & 3\le x \le 4\\
\frac{1}{24}\left(x^4-20x^3+150x^2-500x+625\right) &4\le x\le5
\end{cases}
</math>
 
==Notes==
{{reflist}}
 
==References==
* [[Philip Hall|Hall, Philip]]. (1927) "The Distribution of Means for Samples of Size N Drawn from a Population in which the Variate Takes Values Between 0 and 1, All Such Values Being Equally Probable". ''Biometrika'', Vol. 19, No. 3/4., pp.&nbsp;240&ndash;245.
* Irwin, J.O. (1927) "On the Frequency Distribution of the Means of Samples from a Population Having any Law of Frequency with Finite Moments, with Special Reference to Pearson's Type II". ''Biometrika'', Vol. 19, No. 3/4., pp.&nbsp;225&ndash;239.
 
{{ProbDistributions|continuous-bounded}}
 
{{DEFAULTSORT:Irwin-Hall distribution}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

Latest revision as of 04:07, 12 January 2015

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