Last updated for August 2024

Len Goff

Note that the definition of $\mathbb{E}[X]$ is

This implies that the expectation is equal to $p$ times an expectation according to $F_c$, plus $1-p$ times an expectation according to $F_d$: $$\mathbb{E}[X] = \int_{-\infty}^\infty x\cdot dF(x) = \color{orange}{p} \cdot \int_{-\infty}^\infty x\cdot f(x)\cdot dx + \color{orange}{(1-p)}\cdot \sum_{j} x_j \cdot \pi_j$$