Let X1,X2,...,Xn iid∼ F, a distribution with finite mean µ, and suppose that the momentgenerating function m(t) of X exists for t in an open interval containing 0. Use Theorem 5.2.2 to show that X¯ D...



Let X1,X2,...,Xn


iid∼ F, a distribution with finite mean µ, and suppose that the momentgenerating function m(t) of X exists for t in an open interval containing 0. Use Theorem 5.2.2 to show that X¯


D


→ δµ, the distribution degenerate at µ. (Hint: Write the mgf of X¯


as displayed below, and apply Lemma 5.3.1.


mX¯(t) = (mX (t/n))n =




1+ (t/n) EX + (t


2


/2!n


2


) EX2 + (t


3


/3!n


3


) EX3 +...n


.



Dec 02, 2021
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