| created |
2024-08-29 |
| lastmod |
2024-09-02 |
The moment generating function (MGF) of a real-valued random variable $X$ is the function
$$
M_X(\lambda) := \E[\exp(\lambda X)].
$$
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
$$
\E[\exp(\lambda X)] = \E\left[\sum_{n\geq 0} \frac{\lambda^n X^n}{n!}\right] = \sum_{n\geq 0} \frac{\lambda^n}{n!}\E[X^n],
$$
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]].