Results 1 to 2 of 2

Thread: Order statistics question

  1. #1
    Senior Member
    Joined
    Jan 2008
    From
    Montreal
    Posts
    311
    Awards
    1

    Order statistics question

    I'd like to know if my answers to the following question are correct:

    Let $\displaystyle Y_1 \ Y_2, ..., \ Y_n$ be independent uniformly distributed random variables on the interval $\displaystyle [0, \ \theta]$.


    a) Find the probability distribution function of $\displaystyle Y_{(n)} =\max(Y_1 \ Y_2, ..., \ Y_n)$

    $\displaystyle =n(n-1)[F(y)]^{n-2}[f(y)]^2 + n [F(y)]^{n-1}f'(y)$

    $\displaystyle =n(n-1) \left(\frac{1}{\theta}y \right) ^{n-2} \times \left(\frac{1}{\theta}\right)^2 + n(y)^{n-1}f'(y)$

    $\displaystyle =n(n-1)\left(\frac{1}{\theta}y \right)^{n-2} \times \left(\frac{1}{\theta}\right)^2$


    b) Find the density function of $\displaystyle Y_{(n)}$

    $\displaystyle =n [F(y)]^{n-1}f(y)$

    $\displaystyle n \left(\frac{1}{\theta}y \right)^{n-1} \times \left( \frac{1}{\theta} \right)$


    c) the mean of $\displaystyle Y_{(n)}$

    $\displaystyle \int_0^{\theta} y \times \left[ n(n-1)\left(\frac{1}{\theta}y \right)^{n-2} \times \left(\frac{1}{\theta}\right)^2\right] \ dy$

    I'm not too sure about any of these answers.
    Last edited by lllll; Oct 5th 2008 at 04:00 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    9
    Quote Originally Posted by lllll View Post
    I'd like to know if my answers to the following question are correct:

    Let $\displaystyle Y_1 \ Y_2, ..., \ Y_n$ be independent uniformly distributed random variables on the interval $\displaystyle [0, \ \theta]$.


    a) Find the probability distribution function of $\displaystyle Y_{(n)} =\max(Y_1 \ Y_2, ..., \ Y_n)$

    $\displaystyle =n(n-1)[F(y)]^{n-2}[f(y)]^2 + n [F(y)]^{n-1}f'(y)$

    $\displaystyle =n(n-1) \left(\frac{1}{\theta}y \right) ^{n-2} \times \left(\frac{1}{\theta}\right)^2 + n(y)^{n-1}f'(y)$

    $\displaystyle =n(n-1)\left(\frac{1}{\theta}y \right)^{n-2} \times \left(\frac{1}{\theta}\right)^2$


    b) Find the density function of $\displaystyle Y_{(n)}$

    $\displaystyle =n [F(y)]^{n-1}f(y)$

    $\displaystyle n \left(\frac{1}{\theta}y \right)^{n-1} \times \left( \frac{1}{\theta} \right)$


    c) the mean of $\displaystyle Y_{(n)}$

    $\displaystyle \int_0^{\theta} y \times \left[ n(n-1)\left(\frac{1}{\theta}y \right)^{n-2} \times \left(\frac{1}{\theta}\right)^2\right] \ dy$

    I'm not too sure about any of these answers.
    a) How is a probablity distribution function different to a probability density function. Can you supply the definition you've been given - they would seem to be the same thing to me ......

    b) Correct.

    c) $\displaystyle E(Y_{(n)}) = \int_{0}^{\theta} y \, g(y) \, dy$ where g(y) is the pdf found in (b).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Sep 12th 2011, 02:54 PM
  2. Order statistics
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: Apr 10th 2011, 02:18 PM
  3. Order statistics
    Posted in the Advanced Statistics Forum
    Replies: 10
    Last Post: Apr 4th 2010, 12:43 PM
  4. Order statistics
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: Mar 8th 2009, 10:22 AM
  5. Order Statistics!?
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Feb 2nd 2009, 11:08 AM

Search Tags


/mathhelpforum @mathhelpforum