# MarginalDistribution

MarginalDistribution[dist,k]

represents a univariate marginal distribution of the k coordinate from the multivariate distribution dist.

MarginalDistribution[dist,{k1,k2,}]

represents a multivariate marginal distribution of the {k1,k2,} coordinates.

# Details • The distribution dist can be either a discrete or continuous multivariate distribution.
• For a discrete multivariate distribution dist with PDF , the PDF of MarginalDistribution[dist,{k1,,km}] is given by where ξ={xk1,,xkm}.
• For a continuous multivariate distribution dist with PDF , the PDF of MarginalDistribution[dist,{k1,,km}] is given by where ξ={xk1 ,,xk m }.
• MarginalDistribution can be used with such functions as Mean, CDF, and RandomVariate, etc.

# Examples

open allclose all

## Basic Examples(3)

One-dimensional marginal distributions:

Two-dimensional marginal distributions:

Marginal distributions can be used like any other distribution:

## Scope(34)

### Basic Uses(8)

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 :

### Parametric Distributions(2)

The marginal distributions of many multivariate parametric distributions automatically simplify:

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:

### Nonparametric Distributions(3)

Find marginals of an EmpiricalDistribution:

Cumulative distribution function of the marginal distributions:

Find marginals for a SmoothKernelDistribution:

Find marginals of a HistogramDistribution:

Compare to the histograms of the components:

### Derived Distributions(9)

Find the marginal distribution of a MarginalDistribution:

The second marginal of is the third marginal of :

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 a QuantityDistribution give QuantityDistribution:

Univariate marginal:

Multivariate marginal:

### Automatic Simplifications(12)

#### Discrete Parametric Distributions(3)

Marginals of multivariate DiscreteUniformDistribution again follow a uniform distribution:

Univariate marginals of MultivariatePoissonDistribution follow PoissonDistribution:

Multivariate marginals again follow multivariate Poisson distribution:

Univariate marginals of MultinomialDistribution follow BinomialDistribution:

#### Continuous Parametric Distributions(6)

All univariate marginals of MultinormalDistribution follow NormalDistribution:

Multivariate marginals of MultinormalDistribution are multivariate normal:

Marginals of multivariate UniformDistribution follow uniform distribution:

Multivariate marginals of DirichletDistribution again follow Dirichlet distribution:

Univariate marginals of MultivariateTDistribution follow StudentTDistribution:

Univariate marginals of LogMultinormalDistribution follow LogNormalDistribution:

Multivariate marginals of LogMultinormalDistribution again follow log-multinormal distribution:

#### Derived Distributions(3)

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:

## Applications(5)

Visualize univariate marginal distributions together with a bivariate distribution:

Plot the univariate marginals:

Show the results together:

The city-highway mileage for a midsize car is given by a binormal distribution. Find the city mileage distribution:

Plot the probability density function:

Find the average city mileage:

The male weight and height follow a binormal distribution. Find the height distribution:

Probability density function:

Find the median height for males:

Find the lower quartile:

Express the height distribution in meters:

A fair coin is flipped three times with the objective of getting three tails. Find the join distribution of the number of failures in the form of getting a head on the second or on the third flip:

Probability density function:

Find the average number of failures of each kind:

Find the total number of failures of both kinds:

Simulate the number of failures by getting a head on the second or on the third flip:

A university campus lies completely within twin cities and . On a day there are on average 10 car accidents on campus and the joint distribution of the number of accidents per day in both cities is:

Find the distribution of the number of accidents in each city:

Compare probability density functions:

Simulate the number of accidents per day in city for 30 days:

## Properties & Relations(5)

Use a marginal distribution if an event does not depend on all the variables:

Calculate the event probability:

Find expectations if the function does not involve all the variables:

An -variable multivariate distribution has proper marginal distributions:

Obtain the marginal CDF by taking limits of complementary variables:

Compute the marginal CDF as :

Compute the marginal CDF as :

Obtain the marginal PDF by integrating over complementary variables:

Compute the marginal PDF as :

Compute the marginal PDF as :

Multivariate marginal distributions preserve the correlation between components:

Find the marginal distribution for the first and the third components:

The covariance of the marginal is a submatrix :

For a discrete distribution:

Covariance matrix for :

Define the marginal for the second and the third components:

Covariance for is a submatrix of the covariance of :

## Neat Examples(1)

All six proper marginal PDFs from a trivariate distribution:

Introduced in 2010
(8.0)
|
Updated in 2016
(10.4)