xercise 2.12 Parts 1-3 appeared on the final exam, 2010-11, Term 1.] Let Y,.., Y

Question

xercise 2.12 Parts 1-3 appeared on the final exam, 2010-11, Term 1.] Let Y,.., Yn, be indepen- dent N (, 2) random variables. Their sample variance is i=1 This question explores the properties of S2/o2 and why the tn-1 distribution ap- proaches the standard normal as n oo. Section 2.4.2 argued that S2/02 has the same distribution as X/(n-1), where X ~ 2-1. You may use this result and properties of the x2 distribution without proof. 1. Find E(go/02). 2. Find Var(S2/o2). 3. Let e0 be a fixed constant representing an arbitrarily small "error" (a) As n oo, what is the limiting probability (b) Briefly describe how you would justify the limiting probability (a complete proof is not required). 4. In Section 2.4.3 the sample mean was standardized by its expectation and sam- ple variance and then expanded in equation (2.4) It was shown that this quantity has a tn-1 distribution. As n oo, it that the tn-1 distribution converges to N (0,1). Use the above results to justif this convergence. is known

Explanation / Answer

1) E(X) = n-1 where X follow chi-square with n-1 degree of freedom

hence

E((n-1)s^2/sigma^2) = n-1

hence

(n-1 )E(s^2/sigma^2) = n-1

E(s^2/sigma^2) = 1

2)

Var(X) = 2(n-1)

Var( (n-1) S^2/sigma^2) = 2(n-1)

(n-1)^2 * Var( S^2/sigma^2) = 2(n-1)

Var( S^2/sigma^2) = 2(n-1)/(n-1)^2

Var( S^2/sigma^2) = 2/(n-1)

Please ask rest questions again as per chegg policy we have to answer 1 question I did 2 .

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