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PDF and CDF manipulation

  1. May 28, 2009 #1
    Hello,
    I have this equation:

    [tex]\int_{-\infty}^{\gamma}f_X(x)\,dx+\int_{\gamma}^{\infty}F_Y(x-\gamma)\,f_X(x)\,dx[/tex]

    where [tex]f_X(x)[/tex] and [tex]F_Y(y)[/tex] are the PDF and CDF of the randome variables X and Y, respectively.

    Now the question is: can I write the above equation in the form:

    [tex]1-\int_{0}^{\infty}(...)[/tex]

    Regards
     
  2. jcsd
  3. May 28, 2009 #2

    statdad

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    Try starting with

    [tex]
    \int_{-\infty}^\gamma f_X(x) \, dx = \int_{-\infy}^\infty f_X (x) \, dx - \infty_\gamma^\infty f_X(x) \, dx = 1 - \infty_\gamma^\infty f_X(x) \, dx
    [/tex]
     
  4. May 28, 2009 #3

    D H

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    What statdad meant to say was: Try starting with

    [tex]\int_{-\infty}^{\gamma} f_X(x)\,dx =
    \int_{-\infty}^{\infty} f_X(x)\,dx \;- \,\int_{\gamma}^{\infty}f_X(x\,)dx =
    1 \,- \int_{\gamma}^{\infty}f_X(x)dx[/tex]
     
  5. May 28, 2009 #4

    statdad

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    Yes indeed, but statdad, in his advanced age, was interrupted by some annoying folks at the door and neglected to fix his post. Thanks.
     
  6. May 28, 2009 #5
    Yes, but I want the whole right side be one minus single integral. Is this still doable in some how?
     
  7. May 28, 2009 #6

    statdad

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    Yes: you can write

    [tex]
    1 - \int_\gamma^\infty \left(f_X(x) - F_Y(x-\gamma)f_X(x)\right) \, dx
    [/tex]

    You should be able to take this and write it as

    [tex]
    1 - \int_0^\infty ( \cdots ) \, dx
    [/tex]

    just play around with the integrand.
     
  8. May 28, 2009 #7
    Thank you, but this form is not the one in my mind. I need, if possible, in some how, to eliminate the first term, so that the equation looks like:

    [tex]1-\int_0^{\infty}F_Y(a)\,f_X(a+\gamma)\,da[/tex]​

    Regards
     
  9. May 28, 2009 #8

    statdad

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    Look at the integrand in

    [tex]
    \int_\gamma^\infty \left(f_X(x) - F_Y(x-\gamma)f_X(x)\right) \, dx
    [/tex]

    You should see a very simple way to factor it and then rewrite it in a form more suitable to your desires for this problem.

    Try it - do some work - then post again.
     
  10. May 28, 2009 #9
    I can't see anything that I can do. :shy:
     
  11. May 28, 2009 #10

    statdad

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    Look a little harder. I won't give away the answer.
     
  12. May 29, 2009 #11
    Just give me a hint, I am not strong in probability.
     
  13. May 29, 2009 #12

    statdad

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    Work with the integrand.
     
  14. May 29, 2009 #13
    Are you sure that, we can write [tex]\int_\gamma^\infty \left(f_X(x) - F_Y(x-\gamma)f_X(x)\right) \, dx[/tex] as [tex]\int_0^{\infty}F_Y(a)\,f_X(a+\gamma)\,da[/tex]?
     
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