Expected Value
Definition
The expected value (or expectation) of a random variable $X$ is
- Discrete: $\displaystyle\mathbb{E}[X] = \sum_x x\, p(x)$.
- Continuous: $\displaystyle\mathbb{E}[X] = \int_{-\infty}^{\infty} x\, f(x)\, dx$.
Properties
- Linearity: $\mathbb{E}[aX + bY] = a\,\mathbb{E}[X] + b\,\mathbb{E}[Y]$.
- Law of the unconscious statistician: $\mathbb{E}[g(X)] = \sum_x g(x)\, p(x)$ (discrete) or $\int g(x) f(x)\, dx$ (continuous).