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Math Help - characteristic function of triangular distribution

  1. #1
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    characteristic function of triangular distribution

    Suppose I have:

    f(x)=(1-|x|)1_{(-1,1)}(x)

    I want to show that its characteristic function is:

    \varphi (u)= 2(1-cosu)/u^2

    I let

    E(e^{iux})=\int_{-1}^1e^{iux}(1-|x|).dx
    =\int_{-1}^0e^{iux}(1+x).dx+\int_{0}^1e^{iux}(1-x).dx
    ...
    =\frac{1}{(iu)^2}(e^{-iu}+e^{iu})
    =2(\cos u)/u^2

    May I know what went wrong?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by noob mathematician View Post
    Suppose I have:

    f(x)=(1-|x|)1_{(-1,1)}(x)

    I want to show that its characteristic function is:

    \varphi (u)= 2(1-cosu)/u^2

    I let

    E(e^{iux})=\int_{-1}^1e^{iux}(1-|x|).dx
    =\int_{-1}^0e^{iux}(1+x).dx+\int_{0}^1e^{iux}(1-x).dx
    ...
    =\frac{1}{(iu)^2}(e^{-iu}+e^{iu})
    =2(\cos u)/u^2

    May I know what went wrong?
    The error lies in your ellipsis, show us what you have in that gap and then we may have a chance of seeing what went wrong.

    CB
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  3. #3
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    Oct 2008
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    Hey CB.. Thanks for the pointer.

    The error is indeed found in the (...)
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