Discussion
Suppose \(\underline{X}_{3 \times 1} \sim\) multivariable \(\text{Normal}_{3}(\mu, \Sigma)\) and let
\[ Y = 2X_{1}^{2} - X_{2}^{2} + X_{3}^{2}. \]
Then the probability distribution of Y is :
Answer: D
Y is not \(\chi^{2}\).
A chi-square variable is a sum of squares of independent standard normals and is therefore never negative. \(Y = 2X_{1}^{2} - X_{2}^{2} + X_{3}^{2}\) has a negative coefficient on \(X_{2}^{2}\), so Y takes negative values with positive probability (whenever \(X_{2}^{2}\) is large compared with the other terms), whatever \(\mu\) and \(\Sigma\) are. A variable that can be negative cannot be chi-square.
Even with the sign fixed, the coefficient 2 on \(X_{1}^{2}\) would spoil the chi-square form unless the quadratic form matrix were idempotent with respect to \(\Sigma\). So none of the degrees of freedom offered (2, 3 or 6) applies.
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