a (a) Let Y(1) be the first order statistic of a random sample of size n from a distri- bution that has pdf f(y) = e-(4-

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A A Let Y 1 Be The First Order Statistic Of A Random Sample Of Size N From A Distri Bution That Has Pdf F Y E 4 1 (62.69 KiB) Viewed 23 times

a (a) Let Y(1) be the first order statistic of a random sample of size n from a distri- bution that has pdf f(y) = e-(4-0), o <y <, zero elsewhere. What is the limiting distribution of Zn = n(Y(1) – 6)? e = = = 2 (b) (i) Suppose Xn is the sample mean of a random sample of size n from a distribution that has a Gamma(2, 1/2) (where a = 2, B = ) pdf f(x) = 4.xe-22, 0 SI<, zero elsewhere. Use the Central Limit Theorem to 0 < 0o deduce that the random variables Vn(X, - 1) converges in distribution to - N(0,1/2). (ii) Use the Delta method to find the limiting distribution of the random vari- ables /n(Vĩ, - 1). - . (iii) Use the limiting distribution of part (ii) to find an approximate probability for P(VX36 < 1.15).