Function Repository Resource:

QuadraticFormDistribution

Represent the distribution of a quadratic form with a multivariate normal

Contributed by: Wolfram Research

ResourceFunction["QuadraticFormDistribution"][{a,b,c},{μ,Σ}]

represents the distribution of a quadratic form z.a.z + b.z + c for multivariate normal z.

Details and Options

ResourceFunction["QuadraticFormDistribution"] is the distribution of z.a.z + b.z + c for real-valued p×p symmetric positive definite matrix a, length-p vector b, scalar c, and p-dimensional multivariate normal vector z.

ResourceFunction["QuadraticFormDistribution"] can be used with such functions as Mean and RandomReal.

Examples

Basic Examples (2)

The mean of a quadratic form distribution:

In[1]:=

$Mean[ResourceFunction[ "QuadraticFormDistribution"][{{{Subscript[a, 11], Subscript[a, 12]}, {Subscript[a, 12], Subscript[a, 22]}}, {Subscript[b, 1], Subscript[b, 2]}, c}, {{Subscript[\[Mu], 1], Subscript[\[Mu], 2]}, {{Subscript[\[Sigma], 11]^2, \[Rho]*Subscript[\[Sigma], 11]*Subscript[\[Sigma], 22]}, {\[Rho]*Subscript[\[Sigma], 11]*Subscript[\[Sigma], 22], Subscript[\[Sigma], 22]^2}}}]]$

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The variance of a quadratic form distribution:

In[2]:=

$Variance[ResourceFunction[ "QuadraticFormDistribution"][{{{1, 1/2}, {1/2, 1}}, {Subscript[b, 1], Subscript[b, 2]}, c}, {{Subscript[\[Mu], 1], Subscript[\[Mu], 2]}, {{1, 0}, {0, 1}}}]]$

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Scope (2)

Generate a set of pseudorandom numbers that follow a quadratic form distribution:

In[3]:=

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The skewness and kurtosis of a quadratic form distribution:

In[4]:=

$Skewness[ResourceFunction[ "QuadraticFormDistribution"][{{{1, 1/2}, {1/2, 1}}, {Subscript[b, 1], Subscript[b, 2]}, c}, {{Subscript[\[Mu], 1], Subscript[\[Mu], 2]}, {{1, 0}, {0, 1}}}]]$

Out[4]=

In[5]:=

$Kurtosis[ResourceFunction[ "QuadraticFormDistribution"][{{{1, 1/2}, {1/2, 1}}, {Subscript[b, 1], Subscript[b, 2]}, c}, {{Subscript[\[Mu], 1], Subscript[\[Mu], 2]}, {{1, 0}, {0, 1}}}]]$