> [!NOTE] Definition > Let $X$ be a [[Random Variable|real-valued random variable]]. The **moment generating function** of $X$ is defined as the [[Expectation of Integrable Real-Valued Random Variable|expectation]] $M_{X}(t)=\mathbb{E}[e^{tX}]$for all values of $t$ for which $e^{tX}$ is [[Integrable Real-Valued Random Variable|integrable]].