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Find PDF

  1. Jun 7, 2009 #1
    Hello,

    Suppose that:

    [tex]Z=X_1+X_2+X_1X_2[/tex]

    where [tex]X_i[/tex] for i=1 and 2 are independent and identically distribuited exponential RVs.

    can we find the PDF of Z?

    Regards
     
  2. jcsd
  3. Jun 7, 2009 #2

    EnumaElish

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    You should note that the event {Z = z} is equivalent to {X1 + X2 + X1 X2 = z} or {X1 = (z - X2)/(1 + X2)}. You can use this and use convolutions (example).
     
  4. Jun 7, 2009 #3
    Right, But I want to find the PDF directly, not from differentiating the CDF, if possible. Because these RVs are, actually, not exponentials, but I said so to simplify the problem statement. So I want to avoid the derivative operation, which complicates the whole stituation.

    I say the following:

    let [tex]W=X_1+X_2[/tex] and [tex]Y=X_1X_2[/tex], then [tex]Z=W+Y[/tex]. But we need to evaluate joint PDF of W and Y. Is this approach in the right way?
     
    Last edited: Jun 7, 2009
  5. Jun 7, 2009 #4
    You may want to check the f distribution. The PDF is a bit complicated and I don't have Latex, but you can look it up.
     
    Last edited: Jun 8, 2009
  6. Jun 8, 2009 #5

    EnumaElish

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    You can derive the pdf directly through convolution.
     
  7. Jun 8, 2009 #6
    If we assume that [tex]Y=W+Z[/tex] where [tex]W=X_1X_2[/tex] and [tex]Z=X_1+X_2[/tex], then we need to find the joint PDF [tex]f_{W,Z}(w,z)[/tex], which can be found using Jacobian transformation.

    If we proceed using this, we have:

    [tex]X_1=T_1^{-1}=\frac{W+Z-X_2}{1+X_2}[/tex] and [tex]X_2=T_2^{-1}=\frac{W+Z-X_1}{1+X_1}[/tex]

    Then

    [tex]F_{W,Z}(w,z)=f_{X_1,X_2}(x_1=T_1^{-1},x_2=T_2^{-1})|J|[/tex]

    where [tex]|J|[/tex] is the magnitude of the Jacobian which will be zero in this case!!!!

    Is here anything wrong I did?

    Regards
     
  8. Jun 8, 2009 #7

    EnumaElish

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    You can write X1 as (z - X2)/(1 + X2). Then study the wiki example with normal distribution. How is that example similar to your problem?
     
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