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: