This tag is for questions relating to moment-generating-functions (m.g.f.), which are a way to find moments like the mean$~(μ)~$ and the variance$~(σ^2)~$. Finding an m.g.f. for a discrete random variable involves summation; for continuous random variables, calculus is used.

Since each moment is an expected value, and the definition of expected value involves either a sum (in the discrete case) or an integral (in the continuous case), it would seem that the computation of moments could be tedious. However, there is a single expected value function whose derivatives can produce each of the required moments. This function is called a moment generating function.

Definition: Let $~X~$ be a discrete random variable with probability mass function $~f(x)~$ and support $~S~$. Then: $$M_X(t)=E(e^{tX})=\sum\limits_{x\in S} e^{tx}f(x)$$or, $$M_X(t) = E(e^{tX}) = \int_x e^{tx} f(x) \, \mathrm{d}x$$(the first is discrete, the second continuous) is the moment generating function (or m.g.f.) of $~X~$ as long as the summation is finite for some interval of $~t~$ around $~0~$.

i.e., $~M(t)~$ is the moment generating function of $~X~$ if there is a positive number $~h~$ such that the above summation exists and is finite for $~−h < t < h~$.

Note: There are basically two reasons for which m.g.f.'s are so important.

  • First, the m.g.f. of $~X~$ gives us all moments of $~X~$. That is why it is called the moment generating function.
  • Second, the m.g.f. (if it exists) uniquely determines the distribution. That is, if two random variables have the same m.g.f., then they must have the same distribution.

Thus, if you find the m.g.f. of a random variable, you have indeed determined its distribution.