Let mathbf Y=beginpmatrixy1y2y3endpmatrixsim N3(mathbf 0,mathbf I3) and mathbf A=beginpmatrixfrac13&0&-sqrt2/30&0&0-sqrt2/3&0&frac23endpmatrix. What is the distribution of (mathbf Y'mathbf Amathbf Y)?

Q54Section I (Objective (Q1–Q80))53 of 79

Multivariate Analysis — chi-square distribution via an idempotent form

Let $\mathbf Y=\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}\sim N_3(\mathbf 0,\mathbf I_3)$ and $\mathbf A=\begin{pmatrix}\frac13&0&-\frac{\sqrt2}{3}\\0&0&0\\-\frac{\sqrt2}{3}&0&\frac23\end{pmatrix}$ . What is the distribution of $(\mathbf Y'\mathbf A\mathbf Y)$ ?

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