MGF

The moment generating function (MGF) of a real-valued random variable $X$ is the function

$$M_X(\lambda ):=\mathbb{E}[\exp (\lambda X)].$$

Why is this called the MGF? Because, appropriately enough, it is indeed generated by the moments of $X$. If we write out the Taylor expansion of $e^x$ and use linearity of expectation, we see that

$$\mathbb{E}[\exp (\lambda X)]=\mathbb{E}\left[\sum _{n\le 0}\frac{{\lambda }^n{X}^n}{n!}\right]=\sum _{n\ge 0}\frac{{\lambda }^n}{n!}\mathbb{E}[X^n],$$

so the MGF contains all the moment information of $X$. In particular this implies that the MGF of $x$ is finite iff all moments exist.

Some distributional classes are defined in terms of conditions on the MGF, e.g., sub-Gaussian distributions and sub-exponential distributions.