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

iid Suppose X1, X2, ..., Xn Beta(1, B). Define Yn := n" (1 – X(n)). Show that if r = 1/B, Yn 4 Weibull(shape = B, scale = 1) hint: limn(1+)" = eª 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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Solve the problem attached. Do not reuse a former solution.

Suppose \( X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} \text{Beta}(1, \beta) \). Define \( Y_n := n^r (1 - X_{(n)}) \). Show that if \( r = 1/\beta \), \[ Y_n \overset{d}{\to} \text{Weibull}(\text{shape} = \beta, \text{scale} = 1) \] hint: \(\lim_{n \to \infty}(1 + \frac{a}{n})^n = e^a\) for some constant \( a \).
Transcribed Image Text:Suppose \( X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} \text{Beta}(1, \beta) \). Define \( Y_n := n^r (1 - X_{(n)}) \). Show that if \( r = 1/\beta \), \[ Y_n \overset{d}{\to} \text{Weibull}(\text{shape} = \beta, \text{scale} = 1) \] hint: \(\lim_{n \to \infty}(1 + \frac{a}{n})^n = e^a\) for some constant \( a \).
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