# Casella& Berger problem 8.5 2.c

• Apr 27th 2012, 06:47 PM
camerbabe
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.

• Apr 28th 2012, 06:38 AM
harish21
Re: Casella& Berger problem 8.5 2.c
Quote:

Originally Posted by camerbabe
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.

this might help.Pareto distribution - Wikipedia, the free encyclopedia
• Apr 28th 2012, 07:35 AM
camerbabe
Re: Casella& Berger problem 8.5 2.c
Quote:

Originally Posted by harish21

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.
• Apr 28th 2012, 08:54 AM
harish21
Re: Casella& Berger problem 8.5 2.c
Quote:

Originally Posted by camerbabe
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.
• Apr 28th 2012, 10:37 AM
camerbabe
Re: Casella& Berger problem 8.5 2.c
Quote:

Originally Posted by harish21
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.
• Apr 28th 2012, 03:37 PM
harish21
Re: Casella& Berger problem 8.5 2.c
Quote:

Originally Posted by camerbabe
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.