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Math Help - Expressing a linear function in terms of independent random variables

  1. #1
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    Expressing a linear function in terms of independent random variables

    Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the pdf f(x) = e^{-x} x ranging from 0 to infinity.

    Demonstrate that all linear functions of Y1, Y2,...,Yn such as  \Sigma a_i Y_i can be expressed as a linear function of independent random variables.

    so:

     \Sigma a_i Y_i = a_1Y_1 + a_2Y_2 + ... + a_nY_n = a_1e^{-x_1} + a_2e^{-x_2} +...+a_ne^{-x_n}

    That can't be right though....
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  2. #2
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    Hello,

    Hey... I think you have serious problems with the definitions... f is the pdf, the random variable does not equal e^{-x}
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  3. #3
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    So what would Y equal to? Would I need to do a change of variable?
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    Quote Originally Posted by statmajor View Post
    Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the pdf f(x) = e^{-x} x ranging from 0 to infinity.

    Demonstrate that all linear functions of Y1, Y2,...,Yn such as  \Sigma a_i Y_i can be expressed as a linear function of independent random variables.

    so:

     \Sigma a_i Y_i = a_1Y_1 + a_2Y_2 + ... + a_nY_n = a_1e^{-x_1} + a_2e^{-x_2} +...+a_ne^{-x_n}

    That can't be right though....
    Y_1, Y_2, \dots , Y_n are the order statistics of an Exponential(1) random variable; so W_1 = Y_1, \; W_2 = Y_2 - Y_1, \; W_3 = Y_3 - Y_2, \dots , W_n = Y_n - Y_{n-1} are n independent (exponentially distributed) random variables.
    A proof of this fact can be found in Feller, "An Introduction to Probability Theory and Its Applications, Volume II"; or you may be familiar with the statement that if arrival times are exponentially distributed then the inter-arrival times are independent and exponentially distributed.

    Then
    Y_1 = W_1
    Y_2 = W_2 + W_1
    Y_3 = W_3 + W_2 + W_1
    etc.,
    so the Y's are linear functions of the W's. Hence any linear function of the Y's can be re-written as a linear function of the W's.
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    God, I'm such an idiot. There was another part to this question where it asks me to prove that Z1 = nY1 Z2 = (n-1)(Y2 - Y1) Z3 = (n-2)(Y3-Y2),...,Zn = Yn - Y(n-1)

    so:

    \Sigma a_i Y_i = a_1\frac{Z_1}{n} + a_2(\frac{Z_2}{n-1} + \frac{Z_1}{n})+..+a_n(Z_n + ... + \frac{Z_1}{n})

    Can't believe I didn't realise this sooner. Thanks a lot.

    Last question: is the nth term correct?
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  6. #6
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    The last equation looks consistent with your definition of the Zs.
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  7. #7
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    Thanks for all your help.
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