Moment generating functions (MGFs) can be used to derive probability density functions (PDFs) through their relationship with characteristic functions, which are obtained via Fourier transforms. If the MGF exists in a neighborhood around zero, the characteristic function can be expressed as the MGF evaluated at a complex argument. The inverse Fourier transform of the characteristic function yields the corresponding density function. However, for distributions lacking a density, the process becomes more complex. It is important to note that while MGFs are often thought to have unique properties, characteristic functions are the ones that exhibit uniqueness.
