Discussion
Let \((X_{1}, X_{2}) \sim N_{2}(\mu, \Sigma)\) where \(\Sigma\) is a +ve definite matrix. Then which of the following has a singular normal distribution where \(X_{3} = X_{1} - 2X_{2}\) ?
Answer: A
The singular pair is \(X_{1} + X_{3},\ X_{1} - X_{2}\).
With \(X_{3} = X_{1} - 2X_{2}\):
\[ X_{1} + X_{3} = 2X_{1} - 2X_{2} = 2(X_{1} - X_{2}) \]
so the two components of option A are exact multiples of each other. Their joint normal distribution lives on a line, and its covariance matrix has rank 1: it is singular.
The others are non-singular linear transforms of \((X_{1}, X_{2})\), whose covariance \(\Sigma\) is positive definite: \((X_{1} - X_{3}, X_{2} - X_{3}) = (2X_{2}, 3X_{2} - X_{1})\), \((X_{1}, X_{3})\) and \((X_{2}, X_{3})\) each come from a \(2 \times 2\) matrix with non-zero determinant.
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