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Given a Gaussian distribution $N(\mu_1,\sigma_1^2)$, i would like to choose another mean $\mu_2$ which is $2\sigma_1$ away from $\mu_1$. In this case our new mean $\mu_2=\mu_1\pm 2\sigma_1$.

How do we calculate the new mean($\mu_2$) in multivariate case?

I mean to say, when your multivariate Gaussian distribution is $N(\mu_1,\Sigma_1)$ and my $\Sigma_1$ is symmetric positive definite matrix. i.e $\left[ \begin{array}{cc} \sigma_x^2 & \sigma_{xy} \\ \sigma_{yx} & \sigma_y^2 \end{array} \right]$.

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  • $\begingroup$ I am not sure that I understand even the univariate case. By $\mu_2=\mu_1\pm 2\sigma_1$ you mean that both answers are ok? $\endgroup$ – mpiktas Apr 27 '11 at 10:24
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    $\begingroup$ what is the context of this problem? As it stands now, it is impossible to answer, since in multivariate case the means are vectors and the distance between them is a number, but the covariance matrix is not. $\endgroup$ – mpiktas Apr 27 '11 at 10:26
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In the bivariate case you can substitute the two points ($\mu_2=\mu_1\pm 2\sigma_1$) with an isodensity ellipse: http://www.stat.psu.edu/online/courses/stat505/05_multnorm/06_multnorm_revist.html .

Your $2\sigma_1$ criterion seems a bit arbitrary, but it includes 95.44997% of the random variable. So you may want to use the 95.44997% isodensity ellipse in the bivariate case, too. The principle axes of this rotated ellipse are the eigenvectors of the covariance matrix, see http://web.as.uky.edu/statistics/users/viele/sta601s08/multinorm.pdf . You can generalise this to more than 2 dimensions.

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