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:
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 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:
follows the multivariate distribution dist. 
, 
, ... are independent and follow the distributions 
, 
, ....
can be entered as x Esc dist Esc dist or 
. 
:
, according to a uniform distribution. Compute the expected distance between the two points:
. Find the PDF of the difference between their times of arrival:
be a sum of random variates 
. Distribution of 
may be different from distribution of 
:
may be different from that of 
:
EXAMPLES
Basic Examples 
Scope 