Find the second univariate marginal distribution:

Multivariate marginals depend on the coordinate order given:

Univariate marginals behave as univariate distributions:

Find distribution functions:

Multivariate marginals behave like a multivariate distribution:

Find distribution functions:

Special moments are computed for each univariate marginal distribution:

Compare moments of marginal distributions with the moment of original distribution:

A general multivariate moment cannot typically be found from marginal moments:

Quantile functions can be computed for univariate marginal distributions:

Find the quantile functions:

Or special medians:

Generate random variates from

MarginalDistribution:

Compare the histogram to the plot of the PDF of the marginal distribution:

Estimate distribution parameters:

Define a trivariate probability distribution:

Find marginal distributions:

Find the covariance matrix of

:

The variances of the marginals form the diagonal of the covariance matrix of

:

The marginal distributions of many multivariate parametric distributions automatically simplify:

The univariate marginals follow

BetaDistribution:

The multivariate marginals follow

DirichletDistribution:

In some cases, marginal distributions will not automatically simplify:

Univariate marginals simplify to a

BinomialDistribution:

The multivariate marginals do not simplify:

The resulting marginal can still be used like any other distribution:

Find marginals of an

EmpiricalDistribution:

Cumulative distribution function of the marginal distributions:

Find marginals of a

HistogramDistribution:

Compare to the histograms of the components:

Find the marginal distribution of a

MarginalDistribution:

The second marginal of

is the third marginal of

:

Find marginals of a

CopulaDistribution:

Find marginals of a

TruncatedDistribution:

Probability density function:

Find marginals of a

MixtureDistribution:

Each marginal distribution is the mixture of marginals:

Compare with the marginal distributions of the components:

Marginals of a

MixtureDistribution are mixtures of marginals:

Plot a probability density function for both marginals:

Create a bivariate distribution using marginal distributions:

Compare a PDF of the original distribution

with the

ProductDistribution of marginals:

Compare covariance matrices:

Find marginal distributions of a

TransformedDistribution:

Find marginal distributions of a

ParameterMixtureDistribution:

Visualize the probability density function:

Find marginal distributions of an

OrderDistribution:

Probability density function:

Mean:

Variance:

Marginals of multivariate

DiscreteUniformDistribution again follow a uniform distribution:

Multivariate marginals again follow multivariate Poisson distribution:

Multivariate marginals of

MultinormalDistribution are multivariate normal:

Marginals of multivariate

UniformDistribution follow uniform distribution:

One-dimensional marginals of

DirichletDistribution follow

BetaDistribution:

Multivariate marginals of

DirichletDistribution again follow Dirichlet distribution:

Marginals of

ProductDistribution are the component distributions:

One-dimensional marginal:

A two-dimensional marginal is also defined by

ProductDistribution:

Univariate marginals of a

CopulaDistribution are the marginals used in the specification:

Marginal distributions of a

MixtureDistribution are the mixtures of component marginals: