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In this question , I am unable to follow the hints given by @leonbloy in the second answer . Can somebody please write down a complete solution . I am stuck at the very first step .

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I will not write down a complete solution, but I will get you started. The answer suggested you write $(\overline{X},A)$ as $CX$ for some square matrix $C$. Here I assume $X=(X_1,\dots,X_n)$. Expanding this, you want to fill in the square matrix below so that the following is true: $$\begin{pmatrix} \overline{X}\\X_2-\overline{X} \\ X_3-\overline{X} \\ \vdots \\ X_n-\overline{X}\end{pmatrix} = \begin{pmatrix} * & * & * & \cdots & * \\ * & * & * & \cdots & * \\ * & * & * & \cdots & * \\ \vdots &\vdots & \vdots & \ddots & \vdots \\ * & * & * & \cdots & * \\ \end{pmatrix}\begin{pmatrix} X_1\\X_2 \\ X_3 \\ \vdots \\ X_n\end{pmatrix} $$ So, for instance, $\overline{X}$ should be the dot product of the first row of $C$ with the vector $X$. But you know that $$\overline{X}=\frac{X_1+X_2+\cdots+X_n}{n}$$ so what must the first row of $C$ be? Similarly, you need to figure out how to write each element of the vector $(\overline{X},A)$ as the dot product of a row of $C$ and the vector $X$.

Now the covariance matrix of $(\overline{X},A)$ is given by $CC^T$. So once you have found $C$, compute the $(1,i)$-entry of $CC^T$ to find the covariance of $\overline{X}$ and $A_i$.

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  • $\begingroup$ please give the form of $C$ . I have tried a lot and it still deceives me ! $\endgroup$ – John Apr 30 at 6:18
  • $\begingroup$ Say the first row of $C$ is $(c_{11}, c_{12},\dots,c_{1n})$. Then the dot product of this row with the vector $X$ is $$c_{11}X_1+c_{12}X_2+\cdots + c_{1n}X_n.$$ On the other hand, this dot product should be equal to $$\overline{X} = \frac{1}{n}X_1+\frac{1}{n}X_2+\cdots+\frac{1}{n}X_n$$ so what are the $c_{1i}$'s? Do the same for every row of $C$. $\endgroup$ – kccu Apr 30 at 12:18

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