Expectation

Definition

The expectation denoted as $\mathbb{E}\left\{X\right\}$ of some RV $X\sim f$ is defined as

$$\mathbb{E}\left\{X\right\}={\int }_{-\infty }^{\infty }xf(x)dx$$

and for a particular $h$,

$$\mathbb{E}\left\{h(X)\right\}={\int }_{-\infty }^{\infty }h(x)f(x)dx$$

Theorem (Linearity of Expectation)

The “Average of a Sum” is always equal to the “Sum of the Averages.” This property allows you to break complex expected value problems into simpler, manageable pieces.

For any random variables $X$ and $Y$ (regardless of whether they are independent or dependent), and any constants $a$ and $b$:

1. Additivity: $$\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$$
2. Scaling (Homogeneity): $$\mathbb{E}[aX]=a\cdot \mathbb{E}[X]$$
3. Combined: $$\mathbb{E}[aX+bY]=a\mathbb{E}[X]+b\mathbb{E}[Y]$$