This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)

TransformedDistribution

TransformedDistribution
represents the transformed distribution of expr where the random variable x follows the distribution dist.
TransformedDistribution
represents the transformed distribution of expr where follows the multivariate distribution dist.
TransformedDistribution
represents 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:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
Out[3]=
In[4]:=
Click for copyable input
Out[4]=
 
Transformed distributions can be used like any other distribution:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
Out[3]=
In[4]:=
Click for copyable input
Out[4]=
 
Shift a discrete distribution:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
Out[3]=
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:
Add two discrete independent distributions:
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:
Transform a CopulaDistribution:
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 transformations between SinghMaddalaDistribution and DagumDistribution:
Special transformation of ChiSquareDistribution:
Special transformations of StudentTDistribution:
Special transformation of BetaDistribution:
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:
Compute the CDF for an affine transformation of a Weibull distribution:
Use Assumptions to specify the condition :
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:
New in 8