show that \(\bar X\) and \( (X_i-\bar X)\) are independent.

If all \( X_i\) had same mean \(\mu\) then we transform the random variables \(X_i\); \( [i=1,2,\ldots,n] \) to

\(Y_1=\bar X\)

\(Y_2 =X_2-\bar X \)

\(Y_3 = X_3-\bar X \)

\( \vdots\)

\(Y_n = X_n-\bar X \)

and the Jacobian of the transformation can be shown to not depend on \(X_i\) or \(\bar X\) and is equal to the constant \(n\).(Honestly, I don’t know how do they understand that we need to consider \( Y_1=\bar X\) in lieu of \( Y_1= X_1-\bar X \) and how to compute this Jacobian of the transformation )

But for the above case each \( X_i\) has different mean \(\mu_i\).

What will be the transformation and Jacobian of transformation here?