BinormalDistribution

The bivariate normal distribution over pairs $(x_1, x_2)$.

BinormalDistribution(rho)correlation $\rho$ only — mean $(0,0)$, unit variances.

BinormalDistribution({sigma1, sigma2}, rho)standard deviations and correlation — mean $(0,0)$.

BinormalDistribution({mu1, mu2}, {sigma1, sigma2}, rho)the general form — mean, standard deviations, correlation.

Typed signatures

Type	Package	Applies
((real) -> distribution) & ((list<real>, real) -> distribution) & ((list<real>, list<real>, real) -> distribution)	statistics	always

Domain: Statistics

Details

• PDF at a point $\{x_1, x_2\}$ is the usual bivariate normal density, exact.
• CDF has no elementary closed form here (Owen's T / a two-dimensional integral) — CDF stays unevaluated for BinormalDistribution, a documented gap rather than an approximation.
• Mean is the mean vector $\{\mu_1, \mu_2\}$; Variance is the covariance matrix $\{\{\sigma_1^2, \rho\sigma_1\sigma_2\}, \{\rho\sigma_1\sigma_2, \sigma_2^2\}\}$, both exact.
• RandomVariate draws two independent standard normals and combines them ($x_1=\mu_1+\sigma_1 z_1$, $x_2=\mu_2+\sigma_2(\rho z_1+\sqrt{1-\rho^2}z_2)$) — see RandomVariate for the seeded-PRNG divergence from Wolfram.

Examples

Basic #

$\approx 0.18377629847393068$, cross-checked against wolframscript's N[PDF[BinormalDistribution[1/2],{0,0}]]

live ↗

In$\operatorname{PDF}(\mathrm{BinormalDistribution}(\frac{1}{2}), \bigl\lbrack0, 0\bigr\rbrack)$Out$\frac{\sqrt{3}}{3π}$

live ↗

In$\mathrm{mean}(\mathrm{BinormalDistribution}(\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack1, 1\bigr\rbrack, \frac{1}{2}))$Out$\bigl\lbrack1, 2\bigr\rbrack$

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In$\mathrm{Variance}(\mathrm{BinormalDistribution}(\frac{1}{2}))$Out$\bigl\lbrack\bigl\lbrack1, \frac{1}{2}\bigr\rbrack, \bigl\lbrack\frac{1}{2}, 1\bigr\rbrack\bigr\rbrack$

Possible issues #

Stays unevaluated — no closed form is implemented

live ↗

In$\operatorname{CDF}(\mathrm{BinormalDistribution}(\frac{1}{2}), \bigl\lbrack0, 0\bigr\rbrack)$Out$\operatorname{CDF}(\mathrm{BinormalDistribution}(\frac{1}{2}), \bigl\lbrack0, 0\bigr\rbrack)$

About

Examples	4

See also: NormalDistribution, PDF, Mean, Variance