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Question: stirlings formula which gives approximation for factorials can be derived...

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Stirling’s Formula, which gives approximation for factorials, can be derived using CLT.
(a) Suppose that X1, X2, · · · , Xn is an i.i.d. sample from Exp(1). Show that, for a standard normal

random variable Z, Pleft (rac{_{ar{X}n}-1}{rac{1}{sqrt{n}}}<x ight ) ightarrow Pleft ( Z < x ight )

(b) Show Г(n) уж

by differencing both sides of the approximation in part a. Then set x = 0 to get Stirling’s Formula.

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