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Math Help - Order statistics question

  1. #1
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    Order statistics question

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

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


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

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

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

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


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

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

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


    c) the mean of Y_{(n)}

    \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; October 5th 2008 at 05:00 PM.
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  2. #2
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    Quote Originally Posted by lllll View Post
    I'd like to know if my answers to the following question are correct:

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


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

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

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

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


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

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

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


    c) the mean of Y_{(n)}

    \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) E(Y_{(n)}) = \int_{0}^{\theta} y \, g(y) \, dy where g(y) is the pdf found in (b).
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