# Probability generating function, moment generating function and characteristic function

1. I learned that for any distribution, its characteristic function always exist while its probability generating function and moment generating function may not.

I was wondering if that means characteristic function can be useful anywhere the other two are?

2. Taking into account the existence issue among others, what are some cases, all can be useful? What are some cases, one or two are useful? In other words, I would like to know when to choose which tool solve problems.

For example, probability generating function can be used to solve recurrence relation for discrete distribution. Can the other two also be?

Thanks and regards!

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This might be helpful: math.stackexchange.com/questions/14714/… – Mike Spivey May 5 '11 at 22:46

For a (nonnegative) random variable $X$ having distribution $\mu$ on $[0,\infty)$, it is often convenient to use the Laplace transform: $${\rm E}[e^{ - uX} ] = \int_{[0,\infty )} {e^{ - ux} \mu ({\rm d}x)}, \;\; u \geq 0,$$ which is always defined and bounded from above by $1$.