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

# UniformDistribution

 UniformDistribution represents a continuous uniform statistical distribution giving values between min and max. UniformDistributionrepresents a uniform distribution giving values between 0 and 1. UniformDistributionrepresents a multivariate uniform distribution over the region .
• The probability density for value x in a uniform distribution is constant for , and is zero for or . »
Probability density function of univariate uniform distribution:
Cumulative distribution function of univariate uniform distribution:
Mean and variance of univariate uniform distribution:
Median of univariate uniform distribution:
Probability density function in two dimensions:
Cumulative distribution function in two dimensions:
Mean and variance in two dimensions:
Covariance:
Probability density function of univariate uniform distribution:
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Cumulative distribution function of univariate uniform distribution:
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Mean and variance of univariate uniform distribution:
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Median of univariate uniform distribution:
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Probability density function in two dimensions:
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Cumulative distribution function in two dimensions:
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Mean and variance in two dimensions:
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Covariance:
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 Scope   (10)
Generate a set of random numbers that are uniformly 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:
Distribution parameters estimation for multivariate uniform distribution:
Estimate the distribution parameters from sample data:
Skewness and kurtosis are constant in any dimensions:
The components of multivariate uniform distribution are uncorrelated:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Closed form for symbolic order:
Closed form for symbolic order:
Different mixed moments for a multivariate uniform distribution:
Closed form for symbolic order:
Mixed central moments:
Closed form for symbolic order:
Mixed factorial moments:
Mixed cumulants:
Closed form for symbolic order:
Hazard function:
Hazard function in two dimensions:
Quantile function:
The marginals of multivariate uniform distribution are uniform distributions:
 Applications   (10)
Find the probability that a randomly chosen point is the left part of the interval:
Find the probability that two randomly selected points on a circle create an angle less than :
Generate a uniform distribution of points on a circle:
Obtain a random number from the inverse CDF of a distribution:
The nozzle of a fountain shoots water at speed and angle varying between and with equal probability. Find the expected horizontal distance where water touches the ground:
The phase angle of a sinusoidal signal is uniformly distributed from to . Find the probability that is between and :
Find the probability that the phase angle is at most :
Find the average value of :
Find the probability that is within one standard deviation from the average value:
Two trains arrive at a station independently and stay for 10 minutes. If the arrival times are uniformly distributed, find the probability the two trains will meet at the station within one hour:
The region where the two trains meet:
Shafts are produced with their diameter following uniform distribution over independently of the production of shaft housings, the inner diameter of which follows uniform distribution over . Given that the optimal difference between the diameters is up to , find the probability that a shaft will fit into a housing:
Show the shafts in blue and holes in pink:
The lifetime of a device has uniform distribution. Find the reliability of the device:
The hazard function increasing in time:
Find the reliability of two such devices in a series:
Find the reliability of two such devices in parallel:
Compare the reliability of both systems for and :
Show a distribution function and its histogram in the same plot:
Compare the PDF to its histogram version:
Compare the CDF to its histogram version:
Parameter influence on the CDF for each :
Uniform distribution is closed under scaling and translation:
Assumption of the sign of the scaling factor or explicit numeric value is required:
Truncation:
Relationships to other distributions:
Sum of uniform random variables follows UniformSumDistribution:
For a defined number of variables:
The mean of uniform variables follows BatesDistribution:
Explicitly compute the PDF:
The mean of two uniform random variables follows TriangularDistribution:
Explicitly compute the PDF:
Show using characteristic function:
ExponentialDistribution is the limiting distribution of the where is uniformly distributed:
BetaDistribution is an order distribution of uniformly distributed variables:
The copula distribution of two univariate uniform distributions is a two-dimensional uniform distribution:
UniformDistribution is not defined when either min or max is not a real number:
UniformDistribution is not defined when :
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: