[jamessb]

Yes, this provides good intuition about why it is useful: the PDF of the sum of two random variables is the convolution of the original PDFs. A convolution is awkward to work with, but by the convolution theorem it is a multiplication in the Fourier domain. This immediately suggests that the Fourier transform of a PDF would be a useful thing to work with.

If you don't say that this is what you are doing then it all seems quite mysterious.

[creata]

> the PDF of the sum of two random variables is the convolution of the original PDFs

(Probably obvious to everyone reading, but the variables should be independent.)

[schmidtleonard]

But I'd rather assume the variables are independent and then blame statistics when I get the wrong answer!

[bokenator]

This is a good place to use cumulants. Instead of working with joint characteristic functions, which gets messy, it lets you isolate the effects of correlation into a separate term. The only limitation is that this doesn't work if the moment doesn't exist.