 # Let the joint distribution of (X, Y) be bivariate normal with mean vector ( <mtable r orlovskihmw 2022-07-03 Answered
Let the joint distribution of (X, Y) be bivariate normal with mean vector $\left(\begin{array}{c}0\\ 0\end{array}\right)$ and variance-covariance matrix
$\left(\begin{array}{cc}1& 𝝆\\ 𝝆& 1\end{array}\right)$ , where $-𝟏<𝝆<𝟏$ . Let ${𝚽}_{𝝆}\left(𝟎,𝟎\right)=𝑷\left(𝑿\le 𝟎,𝒀\le 𝟎\right)$ . Then what will be Kendall’s $\tau$ coefficient between X and Y equal to?
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Step 1
Originally, Kendall's tau, also called rank correlation, is a statistical measure that can be applied to a discrete set of observed data.
In the more recent literature about dependency modelling with Copulas which became popular in mathematical finance the following definition of Kendall's tau is given.
Let ${\mathrm{\Phi }}_{\rho }\left(x,y\right),\mathrm{\Phi }\left(x\right),\mathrm{\Phi }\left(y\right)$ be the bivariate and the univariate CDFs of the standard normal distribution. Then the Gaussian Copula is defined as
${C}_{\rho }\left(x,y\right)={\mathrm{\Phi }}_{\rho }\left({\mathrm{\Phi }}^{-1}\left(x\right),{\mathrm{\Phi }}^{-1}\left(y\right)\right)$
Kendall's tau is then defined as
$\begin{array}{rl}{\rho }_{\tau }& =\mathbb{E}\left[\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left[\left(X-\stackrel{~}{X}\right)\left(Y-\stackrel{~}{Y}\right)\right]\right]\\ & =\mathbb{P}\left[\left(X-\stackrel{~}{X}\right)\left(Y-\stackrel{~}{Y}\right)>0\right]-P\left[\left(X-\stackrel{~}{X}\right)\left(Y-\stackrel{~}{Y}\right)<0\right]\phantom{\rule{thinmathspace}{0ex}}.\end{array}$
where (X, Y) is bivariate standard normal, and $\left(\stackrel{~}{X},\stackrel{~}{Y}\right)$ has the same distribution but is independent of (X, Y). It can be shown (see (1) and duplicate) that
${\rho }_{\tau }=4{\int }_{0}^{1}{\int }_{0}^{1}{C}_{\rho }\left(x,y\right)\phantom{\rule{thinmathspace}{0ex}}d{C}_{\rho }\left(x,y\right)-1=\frac{2}{\pi }\mathrm{arcsin}\rho \phantom{\rule{thinmathspace}{0ex}}.$

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