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Math Help - cumulative function

  1. #1
    Junior Member
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    cumulative function

    I have a problem, which I assume is easy to solve, but I don't know exact how to write a coherent argument...

    1. For any distribution function and any a\ge0, we have \int_{-\infty}^\infty [F(x+a)-F(x)]dx=a.

    2. For any compactly supported function \phi:\mathbb{R}\to\mathbb{R} and any probability measure \mu with cumulative function F, one has \int\phi d\mu=-\int\phi '(x)F(x)dx.

    Thanks.
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  2. #2
    MHF Contributor matheagle's Avatar
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    IF you have a density, then

    \int_{-\infty}^\infty [F(x+a)-F(x)]dx=\int_{-\infty}^\infty [P(X\le x+a)-P(X\le x)]dx

    =\int_{-\infty}^\infty P(x<X\le x+a)dx

    =\int_{-\infty}^\infty \int_x^{x+a}f(u)dudx

    =\int_{-\infty}^\infty \int_{u-a}^uf(u)dxdu

    =\int_{-\infty}^\infty af(u)du

    =a\int_{-\infty}^\infty f(u)du=a

    So, it's correct. Next use dF(x) instead of f(x) and switch the order of integration again...

    \int_{-\infty}^\infty \int_x^{x+a}dF(u)dx

    The second one looks like parts.
    Last edited by matheagle; June 8th 2010 at 12:05 AM.
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