# BatesDistribution

represents the distribution of a mean of n random variables uniformly distributed from 0 to 1.

BatesDistribution[n,{min,max}]

represents the distribution of a mean of n random variables uniformly distributed from min to max.

# Background & Context

• BatesDistribution[n,{min,max}] represents a statistical distribution defined over the interval from min to max and parametrized by a positive integer n. The overall shape of the probability density function (PDF) of a Bates distribution varies significantly depending on n and can be uniform, triangular, or unimodal for , , and , respectively. The one-argument form is equivalent to BatesDistribution[n,{0,1}] and is sometimes called the standardized Bates distribution.
• Mathematically, the Bates distribution is defined to be the mean of n statistically independent uniformly-distributed random variables , i.e. XBatesDistribution[n] is equivalent to saying that , where XiUniformDistribution[] for all . The two-argument form BatesDistribution[n,{min,max}] has the same meaning with the exception that UniformDistribution[{min,max}]. One important application of the Bates distribution is in computing, where the standardized Bates distribution with was used historically to generate standard normal variables.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Bates distribution. Distributed[x,BatesDistribution[n,{min,max}]], written more concisely as xBatesDistribution[n,{min,max}], can be used to assert that a random variable x is distributed according to a Bates distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[BatesDistribution[n,{min,max}],x] and CDF[BatesDistribution[n,{min,max}],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a Bates distribution, EstimatedDistribution to estimate a Bates parametric distribution from given data, and FindDistributionParameters to fit data to a Bates distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Bates distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Bates distribution.
• TransformedDistribution can be used to represent a transformed Bates distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Bates distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Bates distributions.
• BatesDistribution is closely related to a number of other distributions. For example, the PDF of a Bates distribution is precisely UniformDistribution and TriangularDistribution for and , respectively, and appears visually similar to the PDF of NormalDistribution for larger values of . BatesDistribution is also closely related to UniformSumDistribution, which represents the sum of statistically independent, uniformly distributed random variables (rather than their mean).

# Examples

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## Basic Examples(4)

Probability density function:

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Cumulative distribution function:

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Mean:

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Variance:

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