Sample mean sample variance Suppose that X1...XN are i.i.d. draws from a distribution with mean and variance 2 . The sample variance S2 is a function of the centered values, Xi X: (See the photo below for the rest)

Sample mean 85 sample variance Suppose that X1, . . . ,Xn are i.i.d. draws from a distribution with mean p and variance 02. The sample variance 52 is a function of the centered values, X;- X. (a) (b) Prove that X is uncorrelated with X1; X (for any single 75), i.e. COV(X, X1- X) : 0. We have learned before that if variables A and B are independent, then A is also independent from any function of B , i.e. A JL 9(3). However, the same does not hold for correlation/ covariance: it's possible to have Corr(A, B) = 0 but Corr(A, 9(3)) 7g 0. Construct a simple example to show that this can occur. Let X1, X2 1'3451' Bernoulli(p) whereif)