1. The moment generating function (MGF) of a random variable $X$ is defined as $$M_X(t) = E[e^{tX}]$$ where $E$ denotes the expected value and $t$ is a real number. 2. The MGF, if it exists in an open interval around $t=0$, uniquely determines the distribution of $X$. 3. The $n$-th moment of $X$ can be found by differentiating the MGF $n$ times with respect to $t$ and evaluating at $t=0$: $$E[X^n] = M_X^{(n)}(0) = \left. \frac{d^n}{dt^n} M_X(t) \right|_{t=0}$$ 4. MGFs are useful because they simplify the calculation of moments and help in proving limit theorems like the Central Limit Theorem. 5. For example, if $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$, its MGF is $$M_X(t) = \exp\left(\mu t + \frac{\sigma^2 t^2}{2}\right)$$ 6. To summarize, MGFs provide a powerful tool to analyze distributions and moments in probability and statistics.