Probability and Statistics

Expectation

Average value of some function $f(x)$ under a probability distribution $p(x)$.

• Discrete $\mathbb{E}[f]={\sum }_{x}p(x)f(x)$
• Continuous $\mathbb{E}[f]=\int p(x)f(x)dx$

In the case of considering expectations of functions of several variables,

$${\mathbb{E}}_x[f(x,y)]$$

denotes the average of the function $f(x,y)$ with respect to the distribution of $x$.