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Determine P(X > x, Y < y)

  1. Jul 27, 2009 #1
    1. The problem statement, all variables and given/known data

    Let U1...Un be independant and uniformly distributed over the unit interval (0,1). Let X be the minimum of U1...Un and Y be the maximum

    a) Determine P(X > x, Y < y). Consider the following cases: 1) 0< x < y < 1 2) 0 < y 1, x < 0
    3) 0 < x < 1, y > 1 4) x < 0, y > 1. 5) all remaining possibilites
    b) Determine the joint CDF of X and Y
    c) using b), determine a joint density funtion of X and Y

    2. Relevant equations



    3. The attempt at a solution

    for a), is the only possible case that can occur is (1)? since it's on the interval (0,1) so X/Y cannot be smaller than 0, and cannot be bigger than 1? And Y also has to be greater than X, since X is the minimum and Y the maximum.

    [tex]\int^1_0 \int^Y_0 dx dy[/tex]

    it doesn't seem right. Any hints?
     
  2. jcsd
  3. Jul 30, 2009 #2
    Anyone?
     
  4. Jul 30, 2009 #3

    HallsofIvy

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    There is something wrong with your problem. You say X and Y are independent. If so, then there is no reason to "Consider the following cases". The probability that X> x is 1- x. The probability that Y< y is y. The probability that X> x and Y<y is (1- x)y.

    And, the CDF of the joint probability if just P(X>x and Y>y)= xy.
     
  5. Jul 30, 2009 #4
    if U1,U2,...,Un are independant, then so are X & Y?
     
  6. Jul 30, 2009 #5

    HallsofIvy

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    Oh, blast! I misread the problem. I thought you were saying X and Y were independent.
     
  7. Jul 30, 2009 #6
    Any ideas?
     
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