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I know in the case of the bivariate normal distribution Kendall's Tau is given by $$ \tau=\frac{2}{\pi}\arcsin({\rho}) $$ where $\rho$ is Pearson's correlation. Can someone given a derivation of this result or provide a reference?

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It is proven as Theorem 3.1 in Fang, Fang, & Kotz, The Meta-elliptical Distributions with Given Marginals Journal of Multivariate Analysis, Elsevier, 2002, 82, 1–16 but that relies on Theorem 2.22 in [K. T. Fang, Kotz, and Ng, "Symmetric Multivariate and Related Distribution," Chapman & Hall, London, 1990.] (to which I do not have access).

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