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.
The Irwin–Hall distribution is the continuous probability distribution for the sum of n independent and identically distributed U(0, 1) random variables:
The probability density function (pdf) for 0 ≤ ≤ --> x ≤ ≤ --> n {\displaystyle 0\leq x\leq n} is given by
where ( x − − --> k ) + {\displaystyle (x-k)_{+}} denotes the positive part of the expression:
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
where the coefficients aj(k,n) may be found from a recurrence relation over k
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.
By the Central Limit Theorem, as n increases, the Irwin–Hall distribution more and more strongly approximates a Normal distribution with mean μ μ --> = n / 2 {\displaystyle \mu =n/2} and variance σ σ --> 2 = n / 12 {\displaystyle \sigma ^{2}=n/12} . To approximate the standard Normal distribution ϕ ϕ --> ( x ) = N ( μ μ --> = 0 , σ σ --> 2 = 1 ) {\displaystyle \phi (x)={\mathcal {N}}(\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:
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:
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.
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).
The Irwin–Hall distribution has an application to beamforming and pattern synthesis in Figure 1 of reference [2][3]
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Former Danish industrial company This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Burmeister & Wain – news · newspapers · books · scholar · JSTOR (April 2020) (Learn how and when to remove this template message) Burmeister & WainIndustryManufacturing, marine engineeringPredecessorBaumgarten & Burmeis…
外埔漁港Waipu Fishing Port外埔漁港位置国家与地区 中華民國(臺灣)位置苗栗縣後龍鎮海埔里经纬度坐标:24°38′59.65″N 120°46′18.03″E / 24.6499028°N 120.7716750°E / 24.6499028; 120.7716750細節运营者南龍區漁會擁有者苗栗縣政府類型第二類漁港[1]僱員甲類582人乙類86人共計:668人數據靠港船隻動力漁船24艘動力漁筏118艘動力舢舨45艘共計:187餘艘年集装箱量466
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