Let X1, , Xn be i i d random variables having the normal distribution with mean and variance 2 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 , variances 2 ...

a Let Y Since X 1 X n are independent we know that Y is independent of X i Since Y ...

Let X1, . . . , Xn be i.i.d. random variables having the normal distribution with mean

Question:

Let X1, . . . , Xn be i.i.d. random variables having the normal with mean μ and variance σ2. Define Xn = 1/n Xi, the sample mean. In this problem, we shall find the conditional of each Xi given n.
a. Show that Xi and Xn have the bivariate normal 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 of Xi given n = n is normal with mean n and variance σ2(1− 1/n). Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...