BUILT-IN MATHEMATICA SYMBOL

BinormalDistribution

represents a bivariate normal distribution with mean

and covariance matrix

.

BinormalDistribution

represents a bivariate normal distribution with zero mean.

BinormalDistribution[

]
represents a bivariate normal distribution with zero mean and covariance matrix

.

MORE INFORMATION

The probability density for vector in a binormal distribution is proportional to .

BinormalDistribution allows to be any real numbers, any positive real numbers, and any number between and .

BinormalDistribution can be used with such functions as Mean, CDF, and RandomVariate.

EXAMPLES

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Basic Examples (4)

Probability density function:

Cumulative distribution function in three dimensions:

Mean and variance:

Covariance:

Probability density function:

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Cumulative distribution function in three dimensions:

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Mean and variance:

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Covariance:

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Scope (7)

Generate a set of pseudorandom vectors that follow a binormal distribution:

Visualize the sample using a histogram:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Goodness-of-fit test:

Skewness and kurtosis are constant vectors:

Correlation of a standard binormal distribution:

Different mixed moments for a standard binormal distribution:

Mixed central moments:

Mixed factorial moments:

Mixed cumulants:

Closed form for a symbolic order:

Hazard function:

Marginal distributions are normal:

Applications (2)

Show a distribution function and its histogram in the same plot:

Compare the PDF to its histogram version:

Compare the CDF to its histogram version:

The average city and highway mileage for midsize cars follows a binormal distribution:

Find the percentage of midsize cars with at least 19 mpg in the city and at least 26 mpg on the highway:

Find the average highway mileage for cars with city mileage of 15 mpg or less:

Show the distribution of city and highway mileage:

Find the mileage, assuming 65% of the driving is done in the city:

Find the average:

Properties & Relations (15)

Equal probability contours for a binormal distribution:

The binormal distribution is closed under affine transformation:

The binormal PDF

satisfies the PDE

:

Hence the CDF

satisfies

and integrates both sides wrt

:

Relationships to other distributions:

The conditional distribution of a binormal distribution is a NormalDistribution:

The conditional distribution differs from marginal distribution when

:

Each bivariate marginal of MultinormalDistribution has binormal distribution:

Binormal distribution is the two-dimensional case of MultinormalDistribution:

Binormal distribution is the limit of a two-dimensional MultivariateTDistribution as

goes to

:

Binormal distribution is related to BeckmannDistribution:

Binormal distribution is related to RiceDistribution:

Binormal distribution is related to RayleighDistribution:

HoytDistribution can be obtained from binormal distribution:

SkewNormalDistribution is a transformation of BinormalDistribution:

The product CopulaDistribution of two normal distributions is a binormal distribution:

A CopulaDistribution with a binormal subkernel and normal marginals is binormal:

Neat Examples (1)

PDFs for different correlations: