Irwin–Hall distribution

Irwin-Hall distribution
Probability density function
Cumulative distribution function
Parameters	n ∈ N₀
Support	$x \in [0,n]$
PDF	${\frac {1}{(n-1)!}}\sum _{{k=0}}^{{\lfloor x\rfloor }}(-1)^{k}{\binom {n}{k}}(x-k)^{{n-1}}$
CDF	${\frac {1}{n!}}\sum _{{k=0}}^{{\lfloor x\rfloor }}(-1)^{k}{\binom {n}{k}}(x-k)^{n}$
Mean	${\frac {n}{2}}$
Median	${\frac {n}{2}}$
Mode	$\begin{cases} \text{any value in } [0,1] & \text{for } n=1 \\ \frac{n}{2} & \text{otherwise} \end{cases}$
Variance	$\frac{n}{12}$
Skewness	0
Ex. kurtosis	$-\tfrac{6}{5n}$
MGF	${\left(\frac{\mathrm{e}^{t}-1}{t}\right)}^n$
CF	${\left(\frac{\mathrm{e}^{it}-1}{it}\right)}^n$

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 independent random variables, each having a uniform distribution.^[1] For this reason it is also known as the uniform sum distribution.

The generation of pseudo-random numbers 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 U(0, 1) random variables:

X = \sum_{k=1}^n U_k.

The probability density function (pdf) is given by

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)

where sgn(x − k) denotes the sign function:

\sgn\left(x-k\right) = \begin{cases} -1 & x < k \\ 0 & x = k \\ 1 & x > k. \end{cases}

Thus the pdf is a spline (piecewise polynomial function) of degree n − 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

f_X(x;n) = \frac{1}{\left(n-1\right)!}\sum_{j=0}^{n-1} a_j(k,n) x^j

where the coefficients a_j(k,n) may be found from a recurrence relation over k

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}

The coefficients are also A188816 in OEIS. The coefficients for the cumulative distribution is A188668.

The mean and variance are n/2 and n/12, respectively.

Special cases

For n = 1, X follows a uniform distribution:

f_X(x)= \begin{cases} 1 & 0\le x \le 1 \\ 0 & \text{otherwise} \end{cases}

For n = 2, X follows a triangular distribution:

f_X(x)= \begin{cases} x & 0\le x \le 1\\ 2-x & 1\le x \le 2 \end{cases}

For n = 3,

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}

For n = 4,

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}

For n = 5,

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}

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+0.5U to address also the case n=1.5 (giving a trapezoidal distribution).

Notes

↑ Johnson, N.L.; Kotz, S.; Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, 2nd Edition, Wiley ISBN 0-471-58494-0(Section 26.9)

References

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. 240–245.
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. 225–239.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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