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Math Help - Help with simulating distributions....

  1. #1
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    Exclamation Help with simulating distributions....

    For each of the following c.d.f F, find a formula for X in terms of U, such that if U~Uniform[0,1], then X has c.d.f F.

    a)
    F(x) =
    0 if 0 x<0
    x if 0<=x<=1
    1 if x>1
    b)
    F(x) =
    0 if 0 x<0
    x^2 if 0<=x<=1
    1 if x>1
    c)
    F(x) =
    0 if 0 x<0
    (x^2)/9 if 0<=x<=3
    1 if x>3

    How do I solve these?
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  2. #2
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    Quote Originally Posted by Sneaky View Post
    For each of the following c.d.f F, find a formula for X in terms of U, such that if U~Uniform[0,1], then X has c.d.f F.

    a)
    F(x) =
    0 if 0 x<0
    x if 0<=x<=1
    1 if x>1
    b)
    F(x) =
    0 if 0 x<0
    x^2 if 0<=x<=1
    1 if x>1
    c)
    F(x) =
    0 if 0 x<0
    (x^2)/9 if 0<=x<=3
    1 if x>3

    How do I solve these?
    You're probably expected to use the probaility integral transform theorem, which states that:

    Suppose that X is a continuous random variable with continuous cdf F(x) and suppose that Y is a continuous standard uniform random variable. Then \displaystyle U = F^{-1}(Y) is a random variable with the same pdf as X.

    There are many references that give the proof and application of this theorem. There have also been questions in this subforum related to this theorem.
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  3. #3
    Senior Member
    Joined
    Sep 2009
    Posts
    299
    are these right?
    a)
    X=U
    b)
    X=sqrt(U)
    c)
    X=3sqrt(U)

    also I dont understand this one
    F( x)=
    0 if x<0
    1/3 if 0<=x<7
    3/4 if 7<=x<=11
    1 if x>= 11
    Last edited by Sneaky; October 30th 2010 at 03:03 PM.
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