View source for Irwin–Hall distribution

{{Short description|Probability distribution}}
{{Probability distribution
  |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 a [[probability distribution]] for a [[random variable]] defined as the 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'''.

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 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) for <math>0\leq x\leq n</math> is given by

:<math>
f_X(x;n)=\frac{1}{(n-1)!}\sum_{k=0}^n (-1)^k{n \choose k} (x-k)_+^{n-1}
</math>

where <math>(x-k)_+</math> denotes the positive part of the expression:

:<math> (x-k)_+ = \begin{cases} 
x-k &  x-k \geq 0 \\
0 &  x-k < 0.\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}{(n-1)!}\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) + (-1)^{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}(-2x^2 + 6x - 3)& 1\le x \le 2\\
\frac{1}{2}(3 - x)^2  & 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}(-3x^3 + 12x^2 - 12x+4)& 1\le x \le 2\\
\frac{1}{6}(3x^3 - 24x^2 +60x-44)  & 2\le x \le 3\\
\frac{1}{6}(4 - x)^3  & 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}(-4x^4 + 20x^3 - 30x^2+20x-5)& 1\le x \le 2\\
\frac{1}{24}(6x^4-60x^3+210x^2-300x+155)  & 2\le x \le 3\\
\frac{1}{24}(-4x^4+60x^3-330x^2+780x-655)  & 3\le x \le 4\\
\frac{1}{24}(5 - x)^4 &4\le x\le5
\end{cases}
</math>

== Approximating a Normal distribution ==
By the [[Central limit theorem|Central Limit Theorem]], as ''n'' increases, the Irwin–Hall distribution more and more strongly approximates a [[Normal distribution]] with mean <math>\mu=n/2</math> and variance <math>\sigma^2=n/12</math>. To approximate the standard Normal distribution <math>\phi(x)=\mathcal{N}(\mu=0, \sigma^2=1)</math>, the Irwin–Hall distribution can be centered by shifting it by its mean of ''n/2'', and scaling the result by the square root of its variance:

:<math>
\phi(x) \overset{n\gg 0}{\approx} \sqrt{\frac{n}{12}} f_X\left(x\sqrt{\frac{n}{12}}+\frac{n}{2};n \right )
</math>
This derivation leads to a computationally simple heuristic that removes the square root, whereby a standard Normal distribution can be approximated with the sum of 12 uniform ''U(0,1)'' draws like so:

:<math>
\sum_{k=1}^{12}U_k -6 \sim f_X(x+6;12) \mathrel{\dot\sim} \phi(x)
</math>

== Similar and related distributions ==
The Irwin–Hall distribution is similar to the [[Bates distribution]], but still featuring only integers as parameter. An extension to real-valued parameters is possible by adding also a random uniform variable with ''N'' − trunc(''N'') as width.
 
==Extensions to the Irwin–Hall distribution==
When using the Irwin–Hall for data fitting purposes one problem is that the IH is not very flexible because the parameter ''n'' needs to be an integer. However, instead of summing ''n'' equal uniform distributions, we could also add e.g. ''U'' + 0.5''U'' to address also the case ''n ='' 1.5 (giving a [[trapezoidal distribution]]).

The Irwin–Hall distribution has an application to beamforming and pattern synthesis in Figure 1 of reference <ref>{{Cite web|url=https://ieeexplore.ieee.org/document/8299700/similar#similar|title = Sidelobe behavior and bandwidth characteristics of distributed antenna arrays|date = January 2018|pages = 1–2}}</ref><ref>https://www.usnc-ursi-archive.org/nrsm/2018/papers/B15-9.pdf {{Bare URL PDF|date=March 2022}}</ref>

== See also ==
* [[Bates distribution]]
* [[Normal distribution]]
* [[Central limit theorem]]
* [[Uniform distribution (continuous)]]
* [[Triangular distribution]]

==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. {{doi|10.1093/biomet/19.3-4.240}} {{JSTOR|2331961}}
* 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. {{doi|10.1093/biomet/19.3-4.225}} {{JSTOR|2331960}}

{{ProbDistributions|continuous-bounded}}

{{DEFAULTSORT:Irwin-Hall distribution}}
[[Category:Continuous distributions]]