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CDF of correlated mixed random variables

  1. Dec 30, 2013 #1
    Hello,

    i m trying to evaluate the following:

    r*x - r*y ≤ g, where r,x,y are nonnegative random variables of different distribution families and g is a constant nonnegative value.

    Then, Pr[r*x - r*y ≤ g] = Pr[r*x ≤ g + r*y] = ∫ Fr x(g + r*y)*fr*y(y) dy, where F(.) and f(.) denote CDF and PDF, respectively.

    The above formula works for independent variables. I m not sure if it works for correlated variables (the above are correlated due to the "r" variable in both "r*x" and "r*y").

    Any help would be useful.

    Thanks
     
  2. jcsd
  3. Dec 30, 2013 #2

    mfb

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    I would express the inequality as ##r(x-y) \leq g## and find the distribution for x-y first, afterwards you just have two variables left.
     
  4. Dec 30, 2013 #3
    However, the distribution of (x-y) is unreachable..Is there an other way to solve the problem ?

    Thanks
     
  5. Dec 30, 2013 #4

    Stephen Tashi

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    I think you'll get a better answer if you explain completely what facts are known about the random variables.
     
  6. Dec 30, 2013 #5
    Ok, assume that x and y are independent and identically distributed variables following a PDF as given bellow:
    fz(z)=Exp[-1/z]/(z^2), where z ε {x,y}

    (Thus, they follow an inverse exponential distribution..)
     
  7. Dec 30, 2013 #6

    mfb

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    I don't understand what is unreachable about the difference of two of those distributions.
    I guess it is not a nice expression, but it is still something you can write down.
     
  8. Dec 30, 2013 #7
    The difference of (x-y) when both of them (x and y) follow the above distribution (i.e., see fz(z)) can not be obtained in closed-form. Moreover, both x and y are i.i.d. and real positive random numbers.

    What else...?
     
  9. Dec 31, 2013 #8

    Stephen Tashi

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    What happens if x > y ?
     
  10. Dec 31, 2013 #9
    Ok, assume that (x-y) ≥ 0, and thus r*(x-y) ≥ 0.

    Thanks
     
  11. Dec 31, 2013 #10

    Stephen Tashi

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    If that is assumed then x and y are not independent.
     
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