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why we use Moment Generating Function in probability theory ? I want to know

1-the main reasons of using (m.g.f.) instead of probability mass function(p.m.f.) and probability density function (p.d.f.).

2-What is the advantage of using Moment Generating Function

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    $\begingroup$ To get an idea of some more advanced use of the moment generating function, have a look at this post: stats.stackexchange.com/questions/191492/… $\endgroup$ – kjetil b halvorsen Feb 6 '17 at 22:51
  • $\begingroup$ Wouldn't an advantage of using mgfs (answering 2) be a reason to use it instead of something else (therefore answering 1)? How does q2 differ from q1? $\endgroup$ – Glen_b Mar 11 '17 at 0:28
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For some specific problems, it saves you a lot of time. For instance, one of the properties of the moment generating function is that $M_{X+Y}(t)=M_X(t)M_Y(t)$ if $X$ and $Y$ are independent. So to figure out the distribution of $X$ and $Y$ you could do it through a convolution of their densities, which is usually long and tedious, or through moment generating functions.

Also, to compute the $k$-th moment of $X$ you can just do $M^{(k)}(0)$, which also can save you time if you need to compute a lot of moments.

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