# 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?.

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

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

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

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