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Math Help - Calculating distribution function of Z = X + Y with X & Y continuous r.v.s

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    Calculating distribution function of Z = X + Y with X & Y continuous r.v.s

    Let X & Y be continuous random variables with joint probability density function f_{X,Y}(x,y). If Z is defined by Z = X + Y, write the distribution function of Z, that is F_Z(z) := \mathbb{P}[X + Y \leq z] as an iterated integral, where the integration is done first with respect to x.

    This is a sample question for a test I have coming up, not sure where to go with it, any help appreciated!
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  2. #2
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    p(X + Y < z) = p(X < z - Y)
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    Had seen that but not sure how to use that fact
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    Integral limits?
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    MHF Contributor matheagle's Avatar
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    are they both uniform (0,1) random variables.
    If so, this is a classic question.
    In tht case you will need to break the region into two parts, from 0 to 1 and from 1 to 2.
    Then you can use geometry.
    Otherwise.........

    P(X+Y\le a) =\int_0^a\int_0^{a-x}f(x,y)dydx =\int_0^a\int_0^{a-y}f(x,y)dxdy
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