For the univariate Normal Distribution, the 68–95–99.7 rule states the percentage of points lying within the intervals defined by the one, two, and three times standard deviation. Or in other words, the probability of a sampled point lying in respective interval is 68%, 95% and 99.7%, respectively.
My question now is: Does this concept translate to the multivariate case? In particular, the Mahalanobis distances measures the distance of a point to the distribution's mean in standard deviations. So can we say that the probability of a randomly sampled point to have a Mahalanobis distance of $< 3$ is 99.7%?
