David Kim

I am studying probability theory and I need to understand moment generating functions (MGFs). The MGF of a random variable $X$ is: $$M_X(t) = E[e^{tX}] = \begin{cases} \sum_x e^{tx} P(X = x) & \text{discrete} \\ \int_{-\infty}^{\infty} e^{tx} f_X(x) \, dx & \text{continuous} \end{cases}$$

I am studying combinatorics and I am fascinated by Catalan numbers. The $n$-th Catalan number is: $$C_n = \frac{1}{n+1} \binom{2n}{n}$$