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Suppose $(X,Y)$ is bi-variate normal distribution with correlation $\rho$ and mean $(0,0)^T$ and both variances 1, and $(X_1,Y_1), (X_2,Y_2),...,(X_n,Y_n)$ is a i.i.d sample from the bi-variate normal distribution. Are $ \sum_{i=1}^n (X_i-\bar{X})^2$ and $\bar{Y}$ independent? $\bar{X}$ and $\bar{Y}$ are the sample means of $X$ and $Y$.

My attempt: Let $x=(X_1,...X_n)^T$ and $y=(Y_1,...,Y_n)^T$, trying to prove $x^TAx$ and $By$ independent where $A={\bf I} -1/n {\bf 1} {\bf 1}^T$ and $B=1/n {\bf 1}^T$. My intuition is they are not independent unless $\rho=0$. Am I wrong?

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