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

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