Let X1,...,Xn be i.i.d. random variables with mean 0 and variance 2 > 0. In class...

Let X1,...,Xn be i.i.d. random variables with mean 0 and variance 2 > 0. In class we have shown a central limit theorem, ¯ Xn /pn )N(0,1), as n!1 , (1) with the assumption E(X1) = 0. Using (1), we now prove the theorem for a more general E(X1)=µ6=0 case. Now suppose X1,...,Xn are i.i.d. random variables with mean µ6= 0 and variance 2. (a) Show that for dummy random variables Yi = Xi µ, E(Yi) = 0 and V ar(Yi)=2. (b) Show that ¯ Yn /pn = ¯ Xnµ /pn . (c) Based on (a) and (b), argue that the central limit theorem holds for µ6= 0.

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orchestra answered 2 years ago

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