Editing Correlation (section)

==Bivariate normal distribution==
If a pair <math>\ (X,Y)\ </math> of random variables follows a [[bivariate normal distribution]], the conditional mean <math>\operatorname{\boldsymbol\mathcal E}(X \mid Y)</math> is a linear function of <math>Y</math>, and the conditional mean <math>\operatorname{\boldsymbol\mathcal E}(Y \mid X)</math> is a linear function of <math>\ X ~.</math> The correlation coefficient <math>\ \rho_{X,Y}\ </math> between <math>\ X\ </math> and <math>\ Y\ ,</math> and the [[Marginal distribution|marginal]] means and variances of <math>\ X\ </math> and <math>\ Y\ </math> determine this linear relationship:

:<math>\operatorname{\boldsymbol\mathcal E}(Y \mid X ) = \operatorname{\boldsymbol\mathcal E}(Y) + \rho_{X,Y} \cdot \sigma_Y \cdot \frac{\ X-\operatorname{\boldsymbol\mathcal E}(X)\ }{ \sigma_X }\ ,</math>

where <math>\operatorname{\boldsymbol\mathcal E}(X)</math> and <math>\operatorname{\boldsymbol\mathcal E}(Y)</math> are the expected values of <math>\ X\ </math> and <math>\ Y\ ,</math> respectively, and <math>\ \sigma_X\ </math> and <math>\ \sigma_Y\ </math> are the standard deviations of <math>\ X\ </math> and <math>\ Y\ ,</math> respectively. 


The empirical correlation <math>r</math> is an [[Estimation|estimate]] of the correlation coefficient <math>\ \rho ~.</math> A distribution estimate for <math>\ \rho\ </math> is given by

: <math display="block">\pi ( \rho \mid r ) =
\frac{\ \Gamma(N)\ }{\ \sqrt{ 2\pi\ } \cdot
\Gamma( N - \tfrac{\ 1\ }{ 2 } )\ } \cdot
\bigl( 1 - r^2 \bigr)^{ \frac{\ N\ - 2\ }{ 2 } } \cdot
\bigl( 1 - \rho^2 \bigr)^{ \frac{\ N - 3\ }{ 2 } } \cdot
\bigl( 1 - r \rho \bigr)^{ - N + \frac{\ 3 \ }{ 2 } } \cdot F_\mathsf{Hyp} \left(\ \tfrac{\ 3\ }{ 2 } , -\tfrac{\ 1\ }{ 2 } ; N - \tfrac{\ 1\ }{ 2 } ; \frac{\ 1 + r \rho\ }{ 2 }\ \right)\ </math>

where <math>\ F_\mathsf{Hyp} \ </math> is the [[Gaussian hypergeometric function]].

This density is both a Bayesian [[posterior probability|posterior]] density and an exact optimal [[confidence distribution]] density.<ref>{{cite journal |last=Taraldsen |first=Gunnar |date=2021 |title=The confidence density for correlation |journal=Sankhya A |volume=85 |pages=600–616 |lang=en |s2cid=244594067 |issn=0976-8378 |doi=10.1007/s13171-021-00267-y |doi-access=free|hdl=11250/3133125 |hdl-access=free }}</ref><ref>{{cite report |last=Taraldsen |first=Gunnar |date=2020 |title=Confidence in correlation |lang=en |type=preprint |doi=10.13140/RG.2.2.23673.49769 |website=researchgate.net |url=http://rgdoi.net/10.13140/RG.2.2.23673.49769}}</ref>