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Suppose X1, X2, ... , X, iid Beta(1, B). Define Y, := n"(1 – X(n)). Show that if r = 1/3, Yn 4 Weibull (shape = B, scale = 1) hint: limn(1+)" = e for some constant a.

Suppose X1, X2, ... , X, iid Beta(1, B). Define Y, := n"(1 – X(n)). Show that if r = 1/3, Yn 4 Weibull (shape = B, scale = 1) hint: limn(1+)" = e for some constant a.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Suppose \( X_1, X_2, \ldots, X_n \sim \text{iid } Beta(1, \beta) \). Define \( Y_n := n^r (1 - X_{(n)}) \). Show that if \( r = 1/\beta \), \[ Y_n \xrightarrow{d} Weibull(shape = \beta, scale = 1) \] Hint: \( \lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a \) for some constant \( a \).
Transcribed Image Text:Suppose \( X_1, X_2, \ldots, X_n \sim \text{iid } Beta(1, \beta) \). Define \( Y_n := n^r (1 - X_{(n)}) \). Show that if \( r = 1/\beta \), \[ Y_n \xrightarrow{d} Weibull(shape = \beta, scale = 1) \] Hint: \( \lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a \) for some constant \( a \).
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