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?
Best Answer
Here's one way to solve the problem. The trick is to substitute
-I*Log[x]
for t in the characteristic function $\varphi[t]$. This turns it into a generating function. Then extract the coefficient of $x^n$ from the resulting series using the SeriesCoefficient built-in. The coefficient of $x^n$ is the value of the pmf f at n. Here's a simple example: