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I would like to recover the probability mass function (pmf) from the characteristic function (CF) of a discrete probability distribution using Mathematica.

Ideally, I'd like to do calculations like this example (not necessarily this simple). To compute the pmf fX+X[x] of the sum of two iid discrete uniform distributions X with support S={1,...,6}, it seems reasonable to try something like this:

 Subscript[\[CurlyPhi], X + X][t_] :=   CharacteristicFunction[DiscreteUniformDistribution[{1, 6}], t]^2  Subscript[f, X + X][x_] :=   InverseFourierTransform[Subscript[\[CurlyPhi], X + X][t], t, x] 

but InverseFourierTransform doesn't recover the pmf from the CF of discrete distributions (it does recover pdfs from the CFs of continuous distributions). Am I using the wrong function? (There are other candidates, but InverseFourier only works on lists of numbers, and InverseZTransform doesn't seem to work here either.) Am I forgetting to set some necessary options to the inverse function? Or is there just no built-in to recover the pmf from the CF of a discrete probability distribution?

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    note that your Characteristic Function is a sum, whereas InverseFourierTransform is an integral2010-09-25
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    @Yaroslav Bulatov: Exactly right. To clarify: I'm only interested in finding out if Mathematica already has a built-in function to recover the pmf of a discrete distribution from its characteristic function. I know it has one to recover the pdf of a continuous distribution from its characteristic function, namely, the InverseFourierTransform function, e.g., InverseFourierTransform[ CharacteristicFunction[NormalDistribution[\\[Mu], \\[Sigma]], t], t, x] produces the pdf $e^{-\frac{(x-\mu )^2}{2 \sigma ^2}} \sqrt{\frac{1}{\sigma ^2}}$, just as expected.2010-09-25
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    Good question...haven't seen it, and it's not in Mathematical Statistics with Mathematica either, if you figure out how to do it you should post a follow-up here2010-09-25

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