Probably worth mentioning the moment-generating function as well, since it's a bit more elementary than the characteristic function and shares many of the properties. It's also more simply related to the probability generating function, you can go from one to the other with a basic change of coordinates (t -> log(x)). I also estimate calculating the moment generating function to be easier in most cases.
In fact most properties of the PGF come from the monent-generating/characteristic function. Including why the second derivative is related to the variance. The second derivative of the moment generating function is the second moment E[X^2]. The second derivative of the logarithm of the MGF is the variance by definition.
The one property that's somewhat unique to the PGF is how composition relates to drawing a randomly-sized sample, which I can see could be useful.
In fact most properties of the PGF come from the monent-generating/characteristic function. Including why the second derivative is related to the variance. The second derivative of the moment generating function is the second moment E[X^2]. The second derivative of the logarithm of the MGF is the variance by definition.
The one property that's somewhat unique to the PGF is how composition relates to drawing a randomly-sized sample, which I can see could be useful.