4

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

kjetil b halvorsen
  • 63,378
  • 26
  • 142
  • 467
zhyan
  • 315
  • 4
  • 11
  • 1
    To get an idea of some more advanced use of the moment generating function, have a look at this post: http://stats.stackexchange.com/questions/191492/how-does-saddlepoint-approximation-work/191781#191781 – kjetil b halvorsen Feb 06 '17 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? – Glen_b Mar 11 '17 at 00:28

1 Answers1

6

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.

Anna SdTC
  • 842
  • 4
  • 10