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Distribution of Maximum of Two Random Variables

  1. Jan 29, 2012 #1
    Hi all,

    I have a random variable (RV):

    [tex]X=\text{max}X_i+X_j[/tex]

    where Xi and Xj are two different RVs from a set of i.i.d N RVs. I need to find the distribution of X. What is the most efficient way?

    Thanks in advance
     
  2. jcsd
  3. Jan 29, 2012 #2

    chiro

    User Avatar
    Science Advisor

    Hey S_David.

    For this problem you can use order statistics and the convolution theorem to get what you want.

    The order statistics is used for getting the Max(X) term and the convolution is used to calculate the distribution for summing independent (but not necessarily indentically distributed) random variables.

    Are you familiar with these?
     
  4. Jan 30, 2012 #3
    I worked with order statistics when I choose one random variable, but in this case I need to pick the maximum two random variables. The problem is that the distribution of each component is very complicated, and I need a way that I can handle this. Actually:

    [tex]X_i=\|h_{1i}\|^2\|h_{2i}\|^2[/tex]

    where each of the components in the multiplication is Chi-square RV with 2L degrees of freedom.

    Thanks
     
  5. Feb 1, 2012 #4

    chiro

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    Science Advisor

    One idea that comes to mind is to use a transformation technique.

    One suggestion I have is to find the pdf/cdf of the square of a chi-squared distribution. Is this what you mean when you use a norm-squared term?

    After this you could use other techniques to find your expression where you multiply two norm-squared terms to get the pdf/cdf of your X_i.

    After that you can use other techniques like order statistics and convolution to do the rest.
     
  6. Feb 1, 2012 #5
    These can get fairly complicated:

    [itex]fX(i),X(j)(u, v) =
    \frac {n!}{(i − 1)!(j − 1 − i)!(n − j)!}
    fX(u)fX(v)[FX(u)]i−1[FX(v)−FX(u)]j−1−i[1−FX(v)]n−j[/itex]

    for [itex]−\infty < u < v < \infty[/itex]

    Maybe you can work backwards from this.
     
    Last edited: Feb 1, 2012
  7. Feb 1, 2012 #6
    The steps are clear in my mind, however, the details are very complicated and the results are very involved.

    I hoped there was an easier way.

    Thanks anyway
     
  8. Feb 1, 2012 #7
    What are these? I am sorry, but I did not get it.

    Thanks
     
  9. Feb 1, 2012 #8
    Sorry. Bad Latex. It should be the joint pdf of X(i),(Xj). This is what you asked for, isn't it?

    [itex]fX(i),X(j)(u, v) =
    \frac {n!}
    {(i − 1)!(j − 1 − i)!(n − j)!}
    fX(u)fX(v)[FX(u)]i−1[FX(v)−FX(u)]j−1−i[1−FX(v)]n−j[/itex]
    for [itex]-\infty< u < v < \infty [/itex]
     
    Last edited: Feb 1, 2012
  10. Feb 1, 2012 #9
    Yeah, I need the joint p.d.f of the summation of the maximum two RVs.
     
  11. Feb 1, 2012 #10
    You quoted it before I could correct another mistake. I think it's OK now.
     
    Last edited: Feb 1, 2012
  12. Feb 1, 2012 #11
  13. Feb 1, 2012 #12
  14. May 7, 2012 #13
    If N is large enough you can use Extreme Value Theory distribution for [tex]\text{max}X_i[/tex] instead order statistics which I think it would simplify the calculations... Good luck!
     
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