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Math Help - p.d.f.s and c.d.f.s of functions

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    p.d.f.s and c.d.f.s of functions

    Let X be a continuous random variable taking values in (a, b) with cumulative distribution function F, strictly increasing
    on (a, b). Show that Y = F(X) has a uniform distribution on (0, 1).

    How would you use a set of computer generated random numbers (assumed to be drawn from a uniform distribution on (0,1) to simulate a random sample from

    f(x) = 1/a . e^(-x/a) x>0

    Not reeally sure at all on this one..

    many thanks in advance
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    Quote Originally Posted by James0502 View Post
    Let X be a continuous random variable taking values in (a, b) with cumulative distribution function F, strictly increasing
    on (a, b). Show that Y = F(X) has a uniform distribution on (0, 1).

    How would you use a set of computer generated random numbers (assumed to be drawn from a uniform distribution on (0,1) to simulate a random sample from

    f(x) = 1/a . e^(-x/a) x>0

    Not reeally sure at all on this one..

    many thanks in advance
    Use F^{-1}(u) as your random sample, where u is a computer-generated (pseudo-)random number drawn from a Uniform(0,1) distribution.

    You need to find a formula for F^{-1}(u) in order to make this practical.

    (This is a general method for generating numbers from arbitrary distributions on a computer.)
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    how would I show the distribution is uniform though?
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    Quote Originally Posted by James0502 View Post
    Let X be a continuous random variable taking values in (a, b) with cumulative distribution function F, strictly increasing
    on (a, b). Show that Y = F(X) has a uniform distribution on (0, 1).

    How would you use a set of computer generated random numbers (assumed to be drawn from a uniform distribution on (0,1) to simulate a random sample from

    f(x) = 1/a . e^(-x/a) x>0

    Not reeally sure at all on this one..

    many thanks in advance
    Also asked here: http://www.mathhelpforum.com/math-he...-question.html
    Last edited by mr fantastic; March 20th 2009 at 07:17 PM.
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    "Let X be a continuous random variable taking values in (a, b) with cumulative distribution function F, strictly increasing
    on (a, b). Show that Y = F(X) has a uniform distribution on (0, 1)."

    The above statement is ture for all pdf's/cdf's, and it is the basis of simulation: generating data for numerous distributions.

    Already mention but here how it works:
    1. generate a random number in (0,1). (Use "=rand()" in MS Excel)
    2. use F-inverse to compute the dist. value you want to generate.

    I don't remember if Normal has an inverse (I think it should). But there is a way of generating Normal-ly distributed values buy above steps. I think they use "characteristic func", which unique for every pdf.

    -O
    Last edited by mr fantastic; February 7th 2009 at 08:00 PM. Reason: Removed link not relevant to question (looks like a signature but isn't)
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    Quote Originally Posted by James0502 View Post
    Let X be a continuous random variable taking values in (a, b) with cumulative distribution function F, strictly increasing
    on (a, b). Show that Y = F(X) has a uniform distribution on (0, 1).

    [snip]
    Since F is strictly increasing, F has an inverse.

    Let  0 \leq y \leq 1. Then

    F(x) \leq y if and only if x \leq <br />
F^{-1}(y)

    so

    P(F(x) \leq y) = P(x \leq F^{-1}(y)) = F(F^{-1}(y)) = y

    I.e., the CDF of Y is Y for 0 \leq Y \leq 1. So Y has a Uniform(0,1) distribution.
    Last edited by awkward; February 7th 2009 at 07:31 PM. Reason: clarification
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