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

TransformedDistribution

 TransformedDistribution represents the transformed distribution of expr where the random variable x follows the distribution dist. TransformedDistributionrepresents the transformed distribution of expr where follows the multivariate distribution dist. TransformedDistributionrepresents a transformed distribution where , , ... are independent and follow the distributions , , ....
• can be entered as x Esc dist Esc dist or .
Simple transformations of random variables:
Transformed distributions can be used like any other distribution:
Shift a discrete distribution:
Simple transformations of random variables:
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Transformed distributions can be used like any other distribution:
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Shift a discrete distribution:
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 Scope   (51)
Scaled distribution:
Compare the PDFs with the probability density function of the original distribution:
Compare medians:
Shifted distribution:
Compare the PDFs:
Generate random numbers following shifted distribution:
Use Assumptions to specify conditions on a parameter in the transformation:
Without assumptions:
Define a nonlinear transformation of a discrete distribution:
Probability density function is defined on integer square roots:
Mean and variance:
Find the distribution of the sum of two different variables:
Probability density function:
Compare the resulting distribution with the summands:
The mean of should be the sum of the means:
Find the distribution of the product:
Probability density function:
Compare all three distributions:
Find skewness and kurtosis:
Use trigonometric functions:
Probability density function:
The domain has been automatically chosen so it is a probability distribution:
Find characteristic function:
Create a piecewise continuous distribution:
Probability density function:
Mean and variance:
Transformation composed of few functions:
Probability density function:
Compare with the original distribution:
Find the distribution of the maximum of two different distributions:
Probability density function:
Cumulative distribution function and survival function:
Hazard function:
Plot all of them:
Find the mean:
Notice it is larger than the means of both original distributions:
Find the distribution of a product of powers of two independent distributions:
Visualize distribution by smooth histogram and histogram based on a random sample:
Scale a bivariate distribution:
Visualize the probability density function:
Create a multivariate distribution given its marginals:
It is the same as using product kernel in copula construction:
Plot the distribution function:
Dimension reducing transformation of a multivariate distribution:
Probability density function:
Mean and variance:
Prove a relation between distributions:
Create a heavy-tail distribution using exponential transformation:
The moments exist only for the orders less than :
Find the distribution of GCD:
Transformation of two identically distributed independent variables:
Probability density function:
Characteristic function:
Cumulant-generating function:
Cumulative distribution function:
Moments:
Central moments:
Cumulants:
Factorial moments:
Create an arbitrary two-dimensional distribution:
Probability density function:
The components are uncorrelated:
Define a bivariate discrete distribution:
Generate a pseudorandom sample:
Density histogram:
Compare means:
Compare standard deviations:
Compare cumulative distribution functions:
Compare probability density functions:
Define a transformed SmoothKernelDistribution:
Compare PDFs:
Complex transformations can be done in steps:
The direct calculation takes too long:
Split the transformation to find the probability density function:
Find a transformation of a MixtureDistribution:
Probability density function:
Compare the PDFs:
The mean is shifted by the same amount as the distribution:
Find a transformation of a ParameterMixtureDistribution:
Cumulative distribution function:
Compare the CDFs:
Standard deviation is scaled by the same factor as the distribution:
Find a transformation of a TruncatedDistribution:
Compare the PDFs:
Find moments:
Find central moments:
Find a transformation of a CensoredDistribution:
Plot the probability density function:
Find a transformation of an OrderDistribution:
Probability density function:
Compare the PDFs:
Mean:
The mean is not the exponent of the mean of the original distribution:
Find a transformation of a MarginalDistribution:
Probability density function:
Probability density function:
Define a transformation of a ProductDistribution:
Probability density function:
Special transformations of NormalDistribution:
Special transformations of ExponentialDistribution:
Special transformations of UniformDistribution:
Special transformation of ChiSquareDistribution:
Special transformations of StudentTDistribution:
Special transformations of BinormalDistribution:
Special transformation of ParetoDistribution:
Special transformations of BernoulliDistribution:
Special transformation of BorelTannerDistribution:
Special transformations of GeometricDistribution:
Special transformations of PoissonDistribution:
Special transformation of PoissonConsulDistribution:
Special transformation of PolyaAeppliDistribution:
Special transformations of SkellamDistribution:
The multinormal distribution is closed under affine transformation:
For specific values:
Multivariate Student distribution is closed under affine transformations:
 Options   (1)
Compute the CDF for an affine transformation of a Weibull distribution:
Use Assumptions to specify the condition :
 Applications   (6)
Two points are chosen randomly and independently from the interval , according to a uniform distribution. Compute the expected distance between the two points:
Two archers shoot at a target. The distance of each shot from the center of the target is uniformly distributed from 0 to 1, independent of the other shot. Find the PDF of the distance of the losing shot from the other:
Romeo and Juliet have a date at a given time, and each, independently, will be late by an amount of time that is exponentially distributed with parameter . Find the PDF of the difference between their times of arrival:
Find the distribution of the distance between the origin and the points placed according to DirichletDistribution on a plane:
Plot the probability density function:
Find the mean distance to the origin:
A driver travels with an average speed of 65 mph for a distance of 120 miles. Assuming the speed has normal distribution with standard deviation of 3 mph and there was no road work, find the distribution of time it takes the driver to cover the distance:
Plot the probability density function:
Find the median travel time in hours:
Concentration-time curve for the circulation of a medication injected in a bloodstream is described by lagged normal distribution:
Compute the first several moments:
Plot the distribution density:
TransformedDistribution uses local names for the variables in the input:
Hence subsequent computations can be done with the original variable name:
The support of the PDF may change under a transformation:
Applying the identity transformation to a distribution leaves it unchanged:
Components of the identity transformation give marginal distributions:
Compute the probability of an event for a transformed distribution:
Compute the expectation of an expression for a transformed distribution:
In particular the extreme cases correspond to Min and Max:
Let be a sum of random variates . Distribution of may be different from distribution of :
Distribution of the sum of two independent identically distributed variates may be different from that of :
Compare distribution densities:
Affine transformations of a normal distribution:
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