# 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

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**