# Moment Generating Function

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

• 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/… Feb 6, 2017 at 22:51
• 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? Mar 11, 2017 at 0:28

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.