A random variable Z is said to have a Gumbel distribution with location parameter µ = 0 and scale parameter β = 1 if Z has cumulative distribution function F(z) = e−e−z, z ∈ R.Let X1,X2,...,Xn be independent and identically distributed exponential random variables with common rate parameter λ = 1. Let Yn = max1≤i≤n Xi. Show that Zn = Yn − lnn converges in distribution to a Gumbel random variable with µ = 0 and β = 1.

and all Xi are independent, therefore the product of probabilities.

Answered: A random variable Z is said to have a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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A random variable Z is said to have a Gumbel distribution with location parameter µ = 0 and scale parameter β = 1 if Z has cumulative distribution function

F(z) = e^−e^−z, z ^∈R.

Let X₁,X₂,...,X_nbe independent and identically distributed exponential random variables with common rate parameter λ = 1. Let Y_n= max_1≤i≤nX_i. Show that Z_n= Y_n− lnn converges in distribution to a Gumbel random variable with µ = 0 and β = 1.

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