Direct proof that X and S2 are independent when sampling fro

Direct proof that X and S^2 are independent when sampling from the N(mu, sigma^2) distribution. Let X1,X2 independent N(mu,sigma^2) random variables. (A random sample of size n = 2.) Show that Y1 = X1 +X2 and Y2 = X2 - X1 are independent. What is the distribution of Y1? What is the distribution of Y2? Show that W1 = 1/2Y1 and W2 = 1/2Y2 are independent. W1 = 1/2 Y1 = 1/2(X1 + X2) = X W2 = 1/2Y2 =1/2(X2 - X1) =1/2X2 -1/2 X1 = X2 -1/2X1 -1/2 X2 =X2 -(X1+X2/2) X2 - X (So X and the n - 1 deviations from the sample mean are independent.) What is the distribution of W1? What is the distribution of W2? Show that since W1 and W2 are independent that W3 = X1 - X is also independent of W1. X1 -X + X2-X = 0 (So X and the first deviation X1 - X are also independent.) Argue that X and S^2 = 1/n - 1 2 sigma i=1(Xi- X)^2 are independent for a random sample of size n = 2 from the N(mu, sigma^2) distribution. What is the distribution of X? What is the distribution of S^2? Develop the same results for n = 3. Develop the same result for a sample of size n.

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