1. The problem is to understand the moment generating function (MGF) and raw moments. 2. The moment generating function $M_X(t)$ of a random variable $X$ is defined as: $$M_X(t) = E[e^{tX}]$$ where $E$ denotes the expected value. 3. The $n$th raw moment of $X$ is the expected value of $X^n$, denoted as: $$\mu'_n = E[X^n]$$ 4. The MGF can be used to find raw moments by differentiating it $n$ times with respect to $t$ and evaluating at $t=0$: $$\mu'_n = M_X^{(n)}(0) = \left. \frac{d^n}{dt^n} M_X(t) \right|_{t=0}$$ 5. This means the first raw moment (mean) is $\mu'_1 = M_X'(0)$, the second raw moment is $\mu'_2 = M_X''(0)$, and so on. 6. Important rules: - The MGF uniquely determines the distribution if it exists in an open interval around zero. - Raw moments provide information about the shape and spread of the distribution. Final answer: The moment generating function $M_X(t) = E[e^{tX}]$ generates raw moments by differentiation: $\mu'_n = M_X^{(n)}(0)$.