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\operatorname{E}[\operatorname{dCov}^2_n(X,Y)] = \frac{n-1}{n^2} \left\{(n-2) \operatorname{dCov}^2(X,Y)+ \operatorname{E}[\|X-X'\|]\,\operatorname{E}[\|Y-Y'\|] \right\} = \frac{n-1}{n^2}\operatorname {E}[\|X-X'\|]\,\operatorname{E}[\|Y-Y'\|].
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For an unbiased estimator of <math>\operatorname{dCov}^2(X,Y)</math> see <ref name=SR2014>Székely & Rizzo (2014)</ref> .
===Distance variance===
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Charles University, Prague.</ref>
If both metric spaces have strong negative type, then <math>\operatorname{dCov}^2(X, Y)= 0</math> iff <math>X, Y</math> are independent.<ref name=Lyonsdcov/>
==Alternative definition==
The definition of distance covariance was motivated by the fact that <math>\operatorname{dCov}(X,Y)</math> is the [[energy distance]] of the joint distribution of <math>\operatorname X, Y </math>and the product of its marginals. But if we want the distance standard deviation to have the same unit of measurement as the random variable <math>\operatorname X </math> (this holds for the classical standard deviation) then we better work with an alternative definition of distance covariance which is simply the square of the definition above, i.e. <math>\operatorname{dCov}^2(X,Y)</math>. In this case we have an unbiased estimator for the distance covariance itself not only for its square <ref name=SR2014>Székely & Rizzo (2014)</ref>. Similarly, in the generalization of distance covariance we can say that <math>\operatorname{dCov}^{2/\alpha}(X, Y; \alpha) </math> is the generalized distance covariance.
==Alternative formulation: Brownian covariance==
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*Pearson, K. (1895). "Note on regression and inheritance in the case of two parents", ''[[Proceedings of the Royal Society]]'', 58, 240–242
*Pearson, K. (1920). "Notes on the history of correlation", ''[[Biometrika]]'', 13, 25–45.
* Székely, G. J. Rizzo, M. L. and Bakirov, N. K. (2007). "Measuring and testing independence by correlation of distances", ''[[The Annals of Statistics]]'', 35/6, 2769–2794. {{doi| 10.1214/009053607000000505}} [http://personal.bgsu.edu/~mrizzo/energy/AOS0283-reprint.pdf Reprint]
* Székely, G. J. and Rizzo, M. L. (2009). "Brownian distance covariance", ''Annals of Applied Statistics'', 3/4, 1233–1303. {{doi| 10.1214/09-AOAS312}} [http://personal.bgsu.edu/~mrizzo/energy/AOAS312.pdf Reprint]
*Kosorok, M. R. (2009) "Discussion of: Brownian Distance Covariance", ''Annals of Applied Statistics'', 3/4, 1270–1278. {{doi|10.1214/09-AOAS312B}} [http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.0822v1.pdf Free access to article]
*Székely, G.J. and Rizzo, M.L. (2014) Partial distance correlation with methods for dissimilarities, The Annals of Statistics, 42/6, 2382-2412.
==External links==
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