---
tag: 0WQVXP7PDB
kind: concept
name: Moment-generating function
aliases: [moment-generating-function, mgf]
msc: ["60E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Moment_generating_function
    locator: lead section
same_as:
  wikidata: Q535587
  wikipedia: Moment generating function
---

## Summary

The moment-generating function of a real random variable $X$ is the function $t \mapsto E[e^{tX}]$. When it is finite near $0$, its derivatives at $0$ are the moments of $X$.
The node [[moment-generating function|0NNRNGKMFM]] gives the precise definition.