Let X1, . . . , Xn be i.i.d. random variables having the normal distribution with mean and variance 2. Define Xn = 1/n

Let X1, . . . , Xn be i.i.d. random variables having the normal distribution with mean μ and variance σ2. Define Xn = 1/n Xi, the sample mean. In this problem, we shall find the conditional distribution of each Xi given n.
a. Show that Xi and Xn have the bivariate normal distribution with both means μ, variances σ2 and σ2/n, and correlation 1/√n. Let Y = Xj. Now show that Y and Xi are independent normals and Xn and Xi are linear combinations of Y and Xi.
b. Show that the conditional distribution of Xi given n = n is normal with mean n and variance σ2(1− 1/n).