Question

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X, be a random sample from N(4,a ), , and let X and S be sample mean and sample 1. Let variance, respectively. In Order to show that and S are independent, tollow the steps below. and show the joint pdf of X,X3,*, X 1-1) Use the change of variable technique = Nx = x - is (п-1)5? п(т-и? f(E,x,) еxp ov2x 2a2 Use Jacobian for n x n variable transformation 1-2) Use the fact that X~N(u,a n) and show that the conditional pdf of X,',X given is п-1 (n-1)s exp 2a2 2T Express the conditional pdf by dividing pdf of X to joint pdf of X,X2,..., X (n-1)S 1-3) Show that the moment generating function(MG F) of X,.., Х, given X is for the conditional distribution of (n-1)S -in-i2 | X (1-2r )l . E еxp ! ,t<= 2 1 [Hint: Notice that g(x,,.,x, x) is a pdf. That is, (n-1)S |X in a multi-integral form using the conditional pdf of 2»*",^, given X E exp Express Then try to consider the integrand as another joint pdf times a constant. Then the answer will be the constant 1-4) From the result of 1-3), how do you argue the independence of X and S [Hint] If conditional MGF docs not depend on a certain statistic, then this is independent of that statistic.

Answer 1

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