This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# BinormalDistribution

 BinormalDistribution represents a bivariate normal distribution with mean and covariance matrix . BinormalDistributionrepresents a bivariate normal distribution with zero mean. BinormalDistribution[]represents a bivariate normal distribution with zero mean and covariance matrix .
• 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 .
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:
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:
The product CopulaDistribution of two normal distributions is a binormal distribution:
A CopulaDistribution with a binormal subkernel and normal marginals is binormal:
PDFs for different correlations:
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