Define a product of two independent continuous distributions:

The PDF is the product of the component PDFs:

Product of discrete distributions:

The PDF is the product of the component PDFs:

Define a product distribution in which three components are repeated:

Probability density function for the four-dimensional product distribution:

Product distribution with both continuous and discrete components:

Draw a random sample from this distribution:

Estimate the distribution parameters for the components using the random sample:

Define a general product distribution with few repeated components:

Compare to a random sample:

Product of multivariate continuous distributions:

Probability density function:

Verify that the integral of the PDF is 1:

Product of multivariate discrete distributions:

Compute the variance of the distribution:

Compare with the values obtained by using a random sample:

Create a bivariate normal distribution with independent components:

Probability density function:

Define a two-dimensional Laplace distribution:

Probability density function:

Mean and variance:

Define product distribution of independent

PoissonDistribution:

Probability density function:

Covariance:

The

MultivariatePoissonDistribution does not have independent components:

The assumptions:

Create the product distribution of two independent examples of

StudentTDistribution:

Generate random sample:

Goodness-of-fit test:

Compute properties with symbolic parameters:

Distribution functions:

Special moments:

Moments with closed forms for symbolic order:

Other moments can be obtained numerically:

Generating functions:

Find product distribution of the marginal distributions:

Probability density function of

:

is a

MultinormalDistribution with a diagonal covariance matrix:

Compare to the product of original distributions:

Create a sample from

and define

SmoothKernelDistribution for this sample:

Compare all three distributions:

Define a product of

EmpiricalDistribution:

Plot the probability density function and cumulative distribution function:

Define a product distribution with

HistogramDistribution:

Probability denstiy function:

Define a product with a

CensoredDistribution:

Compose product distribution from marginals:

Probability density function:

It is the same as for binormal distribution with no correlation:

The components of product distribution are assumed to be independent, hence the original distribution cannot be recovered when

is not zero:

Create the product distribution from a

MixtureDistribution:

Probability density function:

Mean and variance:

Find the product distribution of minimum and maximum

OrderDistribution:

Probability density function:

Plot density function for fixed

:

Define a product distribution of a

ParameterMixtureDistribution:

Product distribution is used as an input for a

TransformedDistribution:

Find the product distribution of a

TransformedDistribution:

Probability density function:

Find the product distribution of a

TruncatedDistribution:

Variance depends on the truncation interval:

Compare the PDF to the product of distributions that are not truncated:

Find the product distribution of a

TruncatedDistribution:

Compare the PDF with the product distribution of two Poisson distributions:

Truncation influences the direction and value of skewness: