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Math Help - sums of independent

  1. #1
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    sums of independent

    Hi, can anyone please help me understand this distribution?
    If X and Y are both uniformly distributed on (0,1), calculate the probabily of X+Y
    Here's what I have:
    fx(a)=fy(a)=1 because both uniformaly distributed.

    Then fx+y(a)=integral (0,1) of fx(a-y)dy
    just found by rearranging x+y=a and fy(a)=1

    Now, it says that fx+y(a)=integral (0,a) dy=a
    My question is how they changed the integration limits

    Why isn't it fx+y(a)=integral (0,1) dy=1?

    Intuitively the first answer makes more sense, but I'm just confused with the math.
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  2. #2
    MHF Contributor matheagle's Avatar
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    the sum of two indep U(0,1) is a triangular density
    that's 0 when S=X+Y is less than 0 or greater than 2.
    It's s on 0<s<1 and it's 2-s on 1<s<2.
    The way to figure that out is via the CDF.
    P(S\le s)=P(X+Y\le s).
    Then draw the unit square of x and y between (0,1) for both
    and find this area WITHOUT integrating, use geometry.
    Last edited by matheagle; April 18th 2009 at 02:00 PM.
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