Question from "Introduction to probability and mathematical statistics" by Bain and Engelhardt
Salutations! I am attempting the problem above.Specifically I am having a problem with part c). I do not understand how the answer at the back of the book was arrived at. Could you please help me figure it out. the answer given is $$0 \, \text{for} \, y \le 1 \\ 1-\frac{1}{y} \, \text{for} \, y\gt 1 \\$$ Thank you!
I found the answer. So just in case anyone struggles with the same problem her is the solution. $$F_{X_{1:n}}(x) = P[X_{1:n}\le x]\\ = 1-P[X_{1:n} \gt x]\\= 1- P[Min(X_{1},X_{2},...X_{n}) \gt x] \\ =1- P[X_{1}\gt x, X_{2} \gt x,...,X_{n} \gt x] \\ =\prod_{i=1}^nP[X_{i:n} \gt x] = 1-[1-F(x)]^n = 1 - (1-(1- \frac{1}{x}))^n \\ = 1 - \frac{1}{x^n} $$ Thus the limit of this goes to 1 as n goes to $\infty.$ Similarly for part c) we have that: $$F_{X_{1:n}^n}(x) = P[X_{1:n}^n\le x]\\ = P[X_{1:n} \le x^{1/n}]\\ = 1 - (x^{1/n})^{-n} = 1- \frac{1}{x} $$
Hence the limit of this expression stays the same when n approaches $\infty$.