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Joint pdf

  1. Dec 1, 2013 #1
    1. The problem statement, all variables and given/known data
    Suppose that the joint pdf of [itex] X [/itex] and [itex]Y[/itex] is
    [tex] f(x,y)= (8/3)xy , 0<x<1,0<y<2, x<y<2x [/tex]
    Compute [tex] P(Y<\sqrt{X}) [/tex]

    2. Relevant equations


    3. The attempt at a solution
    [tex] \int_0^1 \int_x^{\sqrt{x}} (8/3)xy dy dx = (4/3) \int_0^1 x - x^2 dx = 2/9[/tex]
     
    Last edited: Dec 1, 2013
  2. jcsd
  3. Dec 1, 2013 #2

    Ray Vickson

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    Wrong answer: start over. Be very careful about the integration region or regions---always draw a picture first, before writing down your integrals.
     
  4. Dec 1, 2013 #3
    I want to add to this post. Sorry for the double posting. I know this solution I posted above can't be right. I tried switching the order of integration from dydx to dxdy and I get 1/3. One of my classmates suggested doing a bivariate transformation, which I haven't tried but I am a little confused as to why I wouldn't be able to just go ahead and compute this directly.
     
  5. Dec 1, 2013 #4
    Here is the picture I drew. Am I not drawing this right?
     

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  6. Dec 1, 2013 #5

    Ray Vickson

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    Not quite right: for small x > 0 you have the wrong upper limit on y (but it is OK for larger x).
     
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