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Help on to find probability density function

  1. Jun 29, 2011 #1
    hey guys, i am really confused on something.here is the thing:
    i have;


    and here x is uniform distribution on (a,b)
    y is uniform distribution on (c,d)
    x and y independent
    i need to find the probability density function of i but how??????
    actually i dont know how to start!!
  2. jcsd
  3. Jun 29, 2011 #2
    Hi musademirtas! :smile:

    First you will need to know the joint distribution of X and Y. This is easy because of independence:


    Now, to find the pdf, you will need to find the cdf first. That is, for each a, you will want to calculate

    [tex]P\{X+\sqrt{X^2-Y}\leq a\}=\iint_{\{(x,y)~\vert~x+\sqrt{x^2-y}\leq a\}}{f_{X,Y}(x,y)dxdy}[/tex]

    To evaluate this integral, you'll need to know the region [itex]\{(x,y)~\vert~x+\sqrt{x^2-y}\leq a\}[/itex] somewhat better.

    So I suggest you first find out for which tuples the equality holds. That is, for which x and y does it hold that


    (hint: the answer will be a straight line!)
  4. Jul 2, 2011 #3
    hi micromass
    thanks for the help but can you solve it? because i used your help to solve it but i couldnt do it.
  5. Jul 3, 2011 #4
    Well, first you're going to need to write


    in function of y. What do you get for that?
  6. Jul 3, 2011 #5
    ok.i got y=2ax-x^2 then what? how am i going to use this?
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