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Math Help - Casella& Berger problem 8.5 2.c

  1. #1
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    Casella& Berger problem 8.5 2.c

    Hi all,
    I am trying to solve the problem 8.5 part c of Casella & Berger part c. I found the solution online but I couldn't understand why the distribution of log(Xi)-mini(log(Xi) is an exponential distribution where the Xi's follow a pareto distribution.

    Thanks for your help.
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    Re: Casella& Berger problem 8.5 2.c

    Quote Originally Posted by camerbabe View Post
    Hi all,
    I am trying to solve the problem 8.5 part c of Casella & Berger part c. I found the solution online but I couldn't understand why the distribution of log(Xi)-mini(log(Xi) is an exponential distribution where the Xi's follow a pareto distribution.

    Thanks for your help.
    this might help.Pareto distribution - Wikipedia, the free encyclopedia
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    Re: Casella& Berger problem 8.5 2.c

    Quote Originally Posted by harish21 View Post
    In this wikipedia page, xm is a value but in my case, Xm is a ramdom variable representing the minimum of the Xi's. Thanks.
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    MHF Contributor harish21's Avatar
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    Re: Casella& Berger problem 8.5 2.c

    Quote Originally Posted by camerbabe View Post
    In this wikipedia page, xm is a value but in my case, Xm is a ramdom variable representing the minimum of the Xi's. Thanks.
    xm in that page is also the minimum value of X, and is also a random variable(order statistic). look at the definition section of the same page.
    Last edited by harish21; April 28th 2012 at 09:02 AM.
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    Re: Casella& Berger problem 8.5 2.c

    Quote Originally Posted by harish21 View Post
    xm in that page is also the minimum value of X, and is also a random variable(order statistic). look at the definition section of the same page.
    I'm sorry if you find me annoying harish21 but the case I have in hand is about a distribution where the minimum value of the Xi's is v and the the index theta. I am trying to find the distribution of the log(Xi/minXi) for all i. Thanks for your help.
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  6. #6
    MHF Contributor harish21's Avatar
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    Re: Casella& Berger problem 8.5 2.c

    Quote Originally Posted by camerbabe View Post
    I'm sorry if you find me annoying harish21 but the case I have in hand is about a distribution where the minimum value of the Xi's is v and the the index theta. I am trying to find the distribution of the log(Xi/minXi) for all i. Thanks for your help.
    X_1, X_2, \cdots, X_n are identically distributed
    So for Y_i\;=\;\log\dfrac{X_i}{min X_i}, find P(Y_1 \leq y), P(Y_2 \leq y) .. and so on..
    Each Y_i has an exponential(1) distribution.
    Last edited by harish21; April 28th 2012 at 03:48 PM.
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