I have seen the text book derivation where the independence is established through factoring the joint distribution. But has anyone tried to prove that the covariance is zero?.
Let $Z_{i}$ come from a standard normal distribution.
Let $X = \bar{Z}$ be the sample mean and let $$Y = \sum_i\frac{1}{n-1} (Z_{i} -\bar{Z})^2$$ be the sample variance.
Prove that $\operatorname{Cov}(X,Y) = 0$ .
I am getting terms like E($\bar{Z}^3$) in the expansion which is making it very cumbersome to handle...