Let \(X_i\sim N(\mu_i,\sigma^2) \) ; where \( [i=1,2,\ldots,n] \)
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?
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?