Why we need Probability-generating function or MGF? This is a basic question, but I barely find solution about it. I found there are a lot of explanations or examples about def, calculation or properties.  Any master can give me some example about why we need ? Again, the question is WHY we need PGF or MGF(moment generating function). 
In MGF, I learned that MGF can help find PDF, it seems that MGF is better way, because they can do multiplication. Thank you.
 A: Short answer: you don't need the MGF, but it makes certain things easier.  The moment generating function (and its relative the characteristic function, which is actually much more useful because it always exists) provides an alternative way to uncover information about a random variable that is sometimes easier than the usual way.  Here are a couple examples: 


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*Computing moments (mean, variance, skew, etc) is obviously much easier if you have the moment generating function.  This is, after all, why it's called the moment generating function.  Why is this easier?  To compute a moment using a PDF, you have to perform an integral or an infinite sum, which is sometimes very difficult.  With the MGF, you only have to take derivatives, which is comparatively easier.  Take for example a Poisson random variable $X$ with mean $\lambda$: its MGF is $M_X(t) = \exp(\lambda (e^t-1))$.  The mean of $X$ is $M_X^\prime(0)$, the second moment is $M_X^{\prime\prime}(0)$, and so on.  Computing these moments using the PMF of $X$ is more challenging.

*Finding the PDF of a sum of random variables i.e. $Z = X+Y$ is sometimes easier if you know the MGF or characteristic function, since you can simply multiply the MGFs: $M_Z(t) = M_X(t)M_Y(t)$.  You can use this property to prove the central limit theorem, for example. 

