The moment generating function is a generating function for the [[random-variable-moments|moments]] of random variables.
For random variable \(X\) the moment generating function \(M_{X}(t)\) is defined for \(t\in \mathbb{R}\) as \[ M_{X}(t)=\mathbb{E}[e^{tX}]. \]
The function generates moments by differentiation under integration.