This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# UniformSumDistribution

 UniformSumDistribution[n] represents the distribution of a sum of n random variables uniformly distributed from to . UniformSumDistributionrepresents the distribution of a sum of n random variables uniformly distributed from min to max.
• The probability density for value in a uniform sum distribution is proportional to for and zero otherwise.
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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 Scope   (7)
Generate a set of pseudorandom numbers that are uniform sum distributed:
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:
 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 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:
Parameter influence on the CDF for each :
UniformSumDistribution is closed under scaling:
Assumption on the sign of the scaling factor 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:
New in 8