Question: Let X1, . . . , Xn be a random sample where Xi N(, 2 ). Define the random sequence Yn = X n =

Let X1, . . . , Xn be a random sample where Xi N(, 2 ). Define the random sequence Yn = X n = 1 n Xn i=1 Xi . The purpose of this exercise is to demonstrate that Yn converges in distribution

(a) Show that the CDF of Yn is Gn(y) = n(y ) , where is the CDF of the standard normal distribution.

(b)Show that G(y) = limn Gn(y) = 0 if y < , 1/2 if y = , 1, if y >

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