Let X1,...,Xn be i.i.d. r.v.s with mean , variance 2, and skewness . (a) Standardize the
Chapter 6, Problem 11(choose chapter or problem)
Let X1,...,Xn be i.i.d. r.v.s with mean , variance 2, and skewness . (a) Standardize the Xj by letting Zj = (Xj ) . Let Xn and Zn be the sample means of the Xj and Zj , respectively. Show that Zj has the same skewness as Xj , and Zn has the same skewness as Xn. (b) Show that the skewness of the sample mean Xn is / pn. Hint: By (a), we can assume = 0 and 2 = 1 without loss of generality; if the Xj are not standardized initially, then we can standardize them. If (X1 + X2 + + Xn) 3 is expanded out, there are 3 types of terms: terms such as X3 1 , terms such as 3X2 1X2, and terms such as 6X1X2X3. (c) What does the result of (b) say about the distribution of Xn when n is large?
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