The probability density of a point x with respect to a multivariate normal distribution having a mean μ and covariance matrix Σ is given by the equation

Using the sample mean and covariance matrix S as estimates of the mean μ and covariance matrix Σ, respectively, show that the log(prob(x)) is equal to the Mahalanobis distance between a data point x and the sample mean plus a constant that does not depend on x.

prob(X) log prob(x)--log ( (V2n", 1/2)--(x-)-1 (x-)".