Discussion
The Mahalanobis distance between the two Bivariate Normal Population
\[ N_{2}\left(\begin{bmatrix} 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & 4 \end{bmatrix}\right) \text{ and} \]
\[ N_{2}\left(\begin{bmatrix} 2 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & 4 \end{bmatrix}\right) \]
is :
Answer: B
The (squared) Mahalanobis distance is 7/3.
The difference of means is \(\delta = (2 - 1, -2 + 1)' = (1, -1)'\), and the common covariance matrix has
\[ \Sigma^{-1} = \frac{1}{3}\begin{bmatrix} 4 & -1 \\ -1 & 1 \end{bmatrix} \]
(its determinant is \(4 - 1 = 3\)). Then
\[ \begin{aligned} \delta'\Sigma^{-1}\delta &= \frac{1}{3}\left[4(1)^{2} - 2(1)(-1) + (-1)^{2}\right] \\ &= \frac{7}{3} \end{aligned} \]
This \(\Delta^{2}\) is what textbooks call the Mahalanobis distance between the two populations.
Getting the sign of the cross term wrong gives \(\frac{4 - 2 + 1}{3} = 1\), and \(\frac{1}{3}\), \(\frac{10}{3}\) and \(\frac{13}{3}\) do not come from \(\delta'\Sigma^{-1}\delta\) with this \(\delta\) and \(\Sigma\).
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