# UniformSumDistribution

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

UniformSumDistribution[n,{min,max}]

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

# Details • UniformSumDistribution is also known as the IrwinHall distribution.
• The probability density for value in a uniform sum distribution is proportional to for and zero otherwise.
• UniformSumDistribution[n,{min,max}] is equivalent to TransformedDistribution[x1++xn,] with each xi from UniformDistribution[{min,max}].
• UniformSumDistribution allows n to be any positive integer and min and max to be any real numbers with min<max.
• UniformSumDistribution allows min and max to be any quantities with the same unit dimensions and n to be a dimensionless quantity. »
• UniformSumDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• UniformSumDistribution[n,{min,max}] represents a statistical distribution defined over the interval from min to max and parametrized by the positive integer n. The overall shape of the probability density function (PDF) of a uniform sum distribution varies significantly depending on n and can be uniform, triangular, or unimodal with maximum at when , , or , respectively. The one-argument form is equivalent to UniformSumDistribution[n,{0,1}] and is sometimes called the standardized uniform sum distribution. The uniform sum distribution is also known as the IrwinHall distribution.
• The uniform sum distribution is defined to be the sum of n statistically independent, uniformly distributed random variables , i.e. XUniformSumDistribution[n] is equivalent to saying that , where XiUniformDistribution[] for all . The two-argument form UniformSumDistribution[n,{min,max}] has the same meaning, with the exception that XiUniformDistribution[{min,max}]. One important application of the uniform sum distribution is in computing, where the standardized uniform sum distribution with has been used historically to generate standard normal variables. Despite this fact, it should be noted that the is not totally smooth (as is NormalDistribution) because its PDF becomes nonsmooth after taking derivatives. UniformSumDistribution also arises in a number of engineering applications and is particularly useful when modeling the life cycles of various manufactured goods.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a uniform sum distribution. Distributed[x,UniformSumDistribution[n,{min,max}]], written more concisely as xUniformSumDistribution[n,{min,max}], can be used to assert that a random variable x is distributed according to a uniform sum 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[UniformSumDistribution[n,{min,max}],x] and CDF[UniformSumDistribution[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 uniform sum distribution, EstimatedDistribution to estimate a uniform sum parametric distribution from given data, and FindDistributionParameters to fit data to a uniform sum distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic uniform sum distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic uniform sum distribution.
• TransformedDistribution can be used to represent a transformed uniform sum 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 uniform sum distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving uniform sum distributions.
• UniformSumDistribution is closely related to a number of other distributions. For example, the PDF of a uniform sum distribution is precisely UniformDistribution and TriangularDistribution for and , respectively, and appears visually similar to the PDF of NormalDistribution for larger values . (This similarity is due to the fact that tends to NormalDistribution[μ,σ] where μ and σ denote the mean and standard deviation, respectively, of .) UniformSumDistribution is also closely related to BatesDistribution, which represents the mean of statistically independent, uniformly distributed random variables (rather than their sum).

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

## Scope(8)

Generate a sample of pseudorandom numbers from a uniform sum distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness is zero because of the symmetry:

Kurtosis does not depend on the range:

Kurtosis tends to the kurtosis of NormalDistribution:

Different moments with closed forms as functions of parameters:

Hazard function does not have a closed form but can be computed numerically:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Compute quartiles of the displacement:

## Applications(3)

A device has three lifetime stages: A, B, and C. The time spent in each phase follows uniform distribution over ; after phase C, failure occurs. Find the distribution of the time to failure of this device:

Find the mean time to failure:

Find the probability that such a device would be operational for at least 20 hours:

Simulate time to failure for 30 independent devices:

The CDF of the mean of 3 independent uniformly distributed random variables:

The CDF can also be derived from the UniformSumDistribution:

Show that they are the same:

Get the corresponding PDFs by taking derivatives:

Generate random numbers from its definition:

Compare the data histogram with the PDF:

Find parameters min and max to approximate standard normal distribution using the method of moments:

Compare densities of the standard normal distribution and its approximation:

## Properties & Relations(6)

UniformSumDistribution is closed under scaling:

Assumption on the sign of scale or numeric value is required:

Relationships to other distributions: Sum of n uniform random variables follows UniformSumDistribution:

The mean of n uniform variables follows BatesDistribution:

The mean of two uniform variables follows TriangularDistribution:

UniformSumDistribution illustrates the central limit theorem:

## Neat Examples(1)

PDFs for different n values with CDF contours: