# Expectation

If a $X$ is a random variable, then a function $g(x)$ is also a random variable. We will touch on this again we talk about [How do distributions transform under a change of variables?](distributions/change-of-variables). 

The **expected value** of a function $g(x)$, which may just be $x$ itself or a component of $x$, is defined by

$$
\mathbb{E}[g(x)] := \int g(x) p_X(x) dx
$$

```{admonition} Synonymous terms: 
Expected value, expectation, mean, or average, or first moment .
```

Note in physics, one would often write $\langle g \rangle$ for the expected value of $g$. 

Note, sometimes one writes $\mathbb{E}_{p_X}$ to make the distribution $p_X$ more explicit. 

## Expectations with emperical data

If $\{x_i\}_{i=1}^N$ is a dataset (emperical distribution) with independent and identically distributed (iid) $x_i \sim p_X$, then one can estimate the expectation with the **sample mean**

$$
\mathbb{E}[g(x)] \approx \frac{1}{N} \sum_{i=1}^N g(x_i)
$$