This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 BUILT-IN MATHEMATICA SYMBOL See Also »|More About »

# MarginalDistribution

 MarginalDistributionrepresents a univariate marginal distribution of the k coordinate from the multivariate distribution dist. MarginalDistributionrepresents a multivariate marginal distribution of the coordinates.
• The distribution dist can be either a discrete or continuous multivariate distribution.
• For a discrete multivariate distribution dist with PDF , the PDF of MarginalDistribution is given by where .
• For a continuous multivariate distribution dist with PDF , the PDF of MarginalDistribution is given by where .
One-dimensional marginal distributions:
Two-dimensional marginal distributions:
Marginal distributions can be used like any other distribution:
One-dimensional marginal distributions:
 Out[1]=
 Out[2]=

Two-dimensional marginal distributions:
 Out[1]=
 Out[2]=

Marginal distributions can be used like any other distribution:
 Out[1]=
 Out[2]=
 Out[3]=
 Scope   (32)
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 for a SmoothKernelDistribution:
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:
Univariate marginals of MultinomialDistribution follow BinomialDistribution:
All univariate marginals of MultinormalDistribution follow NormalDistribution:
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:
Univariate marginals of MultivariateTDistribution follow StudentTDistribution:
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:
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:
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 :
All six proper marginal PDFs from a trivariate distribution:
New in 8