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

# BatesDistribution

 BatesDistribution[n] represents the distribution of a mean of n random variables uniformly distributed from to . BatesDistributionrepresents the distribution of a mean 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.
• BatesDistribution allows n to be any positive integer and min and max to be any real numbers with .
Probability density function:
Cumulative distribution function:
Mean:
Variance:
Probability density function:
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Cumulative distribution function:
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Mean:
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Variance:
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 Scope   (7)
Generate a set of pseudorandom numbers that are Bates distributed:
Compare the 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:
Kurtosis does not depend on the domain:
In the limit, kurtosis is the same as kurtosis of NormalDistribution:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Closed form for symbolic order:
Closed form for symbolic order:
Hazard function:
Quantile function:
 Applications   (2)
The mean of a uniform distribution follows a Bates distribution:
A device has 3 lifetime phases: A, B, and C. The time spent in each phase follows uniform distribution over , and after phase C, failure occurs. Find the distribution of the average time spent in the same phase:
Find the mean average time spent in the same phase:
Find the probability that on average the device will stay in the same stage for more than 4 hours:
Simulate the average time spent in the same phase for 30 such devices:
Parameter influence on the CDF for each :
Bates distribution is closed under translation and scaling:
Scaling with a negative factor:
The default domain is interval :
Relationships to other distributions:
The distribution of the mean of n uniformly distributed variables follows the Bates distribution:
The triangular distribution is a special case of the Bates distribution:
New in 8