Irwin–Hall distribution: Information from Answers.com

In probability theory and statistics, the Irwin–Hall distribution, named after Joseph Oscar Irwin and Philip Hall, is a continuous probability distribution of the sum of n i.i.d. U(0, 1) random variables:

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

For this reason it is also known as the uniform sum distribution. The probability density function (pdf) is given by

$f_X(x)=\frac{1}{2\left(n-1\right)!}\sum_{k=0}^{n}\left(-1\right)^k{n \choose k}\left(x-k\right)^{n-1}\sgn(x-k)$

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

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

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

$a_j(k,n)=\begin{cases}I(j=n-1)&k=0\\ a_j(k-1,n) + \left(-1\right)^{n+k-j-1}{n\choose k}{{n-1}\choose j}k^{n-j-1} &k>0\end{cases}$

The 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,

$f_X(x)= \begin{cases} \frac{1}{24}x^4 & 0\le x \le 1\\ \frac{1}{24}\left(-4x^4 + 20x^3 - 30x^2+20x-5 \right)& 1\le x \le 2\\ \frac{1}{24}\left(6x^4-60x^3+210x^2-300x+155 \right) & 2\le x \le 3\\ \frac{1}{24}\left(-4x^4+60x^3-330x^2+780x-655 \right) & 3\le x \le 4\\ \frac{1}{24}\left(x^4-20x^3+150x^2-500x+625\right) &4\le x\le5 \end{cases}$

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

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

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