Irwin–Hall Distribution
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In
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the Irwin–Hall distribution, named after
Joseph Oscar Irwin Joseph Oscar Irwin (17 December 1898 – 27 July 1982) was a British statistician who advanced the use of statistical methods in biological assay and other fields of laboratory medicine. Irwin's grasp of modern mathematical statistics distin ...
and
Philip Hall Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups. Biography He was educated first at Christ's Hospital, where he won the Thom ...
, is a
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
for a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
defined as the sum of a number of
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
random variables, each having a uniform distribution. For this reason it is also known as the uniform sum distribution. The generation of
pseudo-random number A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Pseudorandom number generators are often used in computer programming, as tradi ...
s having an approximately
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
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 In probability and business statistics, the Bates distribution, named after Grace Bates, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval. This dist ...
, 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 In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
for the sum of ''n''
independent and identically distributed Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
''U''(0, 1) random variables: : X = \sum_^n U_k. The
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
(pdf) for 0\leq x\leq n is given by : f_X(x;n)=\frac\sum_^n (-1)^k (x-k)_+^ where (x-k)_+ denotes the positive part of the expression: : (x-k)_+ = \begin x-k & x-k \geq 0 \\ 0 & x-k < 0.\end Since k is an integer, we have that (x-k)_+ = (x- k) if and only if k \leq \lfloor x \rfloor. Hence, a completely equivalent expression of the pdf for 0 \leq x \leq n is given by : f_X(x; n) = \frac \cdot \sum_^ (-1)^k \binom (x-k)^. 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\sum_^ a_j(k,n) x^j where the coefficients ''a''''j''(''k'',''n'') may be found from a
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
over ''k'' : a_j(k,n)=\begin 1&k=0, j=n-1\\ 0&k=0, j0\end The coefficients are als
A188816
in
OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...
. The coefficients for the cumulative distribution i
A188668
The
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
and
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
are ''n''/2 and ''n''/12, respectively.


Special cases

* For ''n'' = 1, ''X'' follows a uniform distribution: :: f_X(x)= \begin 1 & 0\le x \le 1 \\ 0 & \text \end * For ''n'' = 2, ''X'' follows a
triangular distribution In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit ''a'', upper limit ''b'', and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''. ...
: :: f_X(x)= \begin x & 0\le x \le 1\\ 2-x & 1\le x \le 2 \end * For ''n'' = 3, :: f_X(x)= \begin \fracx^2 & 0\le x \le 1\\ \frac(-2x^2 + 6x - 3)& 1\le x \le 2\\ \frac(3 - x)^2 & 2\le x \le 3 \end * For ''n'' = 4, :: f_X(x)= \begin \fracx^3 & 0\le x \le 1\\ \frac(-3x^3 + 12x^2 - 12x+4)& 1\le x \le 2\\ \frac(3x^3 - 24x^2 +60x-44) & 2\le x \le 3\\ \frac(4 - x)^3 & 3\le x \le 4 \end * For ''n'' = 5, :: f_X(x)= \begin \fracx^4 & 0\le x \le 1\\ \frac(-4x^4 + 20x^3 - 30x^2+20x-5)& 1\le x \le 2\\ \frac(6x^4-60x^3+210x^2-300x+155) & 2\le x \le 3\\ \frac(-4x^4+60x^3-330x^2+780x-655) & 3\le x \le 4\\ \frac(5 - x)^4 &4\le x\le5 \end


Approximating a Normal distribution

By the
Central Limit Theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, as ''n'' increases, the Irwin–Hall distribution more and more strongly approximates a
Normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
with mean \mu=n/2 and variance \sigma^2=n/12. To approximate the standard Normal distribution \phi(x)=\mathcal(\mu=0, \sigma^2=1), 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: : \phi(x) \overset \sqrt f_X\left(x\sqrt+\frac;n \right ) 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: : \sum_^U_k -6 \sim f_X(x+6;12) \mathrel \phi(x)


Similar and related distributions

The Irwin–Hall distribution is similar to the
Bates distribution In probability and business statistics, the Bates distribution, named after Grace Bates, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval. This dist ...
, 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 In probability theory and statistics, the trapezoidal distribution is a continuous probability distribution whose probability density function graph resembles a trapezoid. Likewise, trapezoidal distributions also roughly resemble mesas or plateau ...
). The Irwin–Hall distribution has an application to beamforming and pattern synthesis as shown in Figure 1 of reference.


See also

*
Bates distribution In probability and business statistics, the Bates distribution, named after Grace Bates, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval. This dist ...
*
Normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
*
Central limit theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
*
Uniform distribution (continuous) In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that li ...
*
Triangular distribution In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit ''a'', upper limit ''b'', and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''. ...


Notes


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. {{DEFAULTSORT:Irwin-Hall distribution Continuous distributions