Direct proof that \(\bar{X}\) and \(S^2\) are independent when sampling from the \(N(\mu, \sigma^2)\) distribution. Let \(X_1, X_2\) be independent \(N(\mu, \sigma^2)\) random variables.

  1. Show that \(Y_1 = X_1 + X_2\) and \(Y_2 = X_2 - X_1\) are independent.
  2. What is the distribution of \(Y_1?\) What is the distribution of \(Y_2?\)
  3. Show that \(W_1 = \frac{1}{2}Y_1\) and \(W_2 = \frac{1}{2}Y_2\)
  4. What is the distribution of \(W_1?\) What is the distribution of \(W_2\)

Solution

  1. To show that \(Y_1\) and \(Y_2\) are independent we will want to show that the product of the marginal distributions results in the joint distribution of these. So first lets find the joint distribution of \(Y_!\) and \(Y_2\):

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