Regarding Proof of the independence of the sample mean and sample variance

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 .

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 $$A_i$$.
• please give the form of $C$ . I have tried a lot and it still deceives me ! – John Apr 30 at 6:18
• 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 $C$. – kccu Apr 30 at 12:18