A particularly useful class of special expectation values are the moments. The \( n \)-th moment of the PDF \( p \) is defined as follows:

$$ \langle x^n\rangle \equiv \int\! x^n p(x)\,dx $$

The zero-th moment \( \langle 1\rangle \) is just the normalization condition of \( p \). The first moment, \( \langle x\rangle \), is called the mean of \( p \) and often denoted by the letter \( \mu \):

$$ \langle x\rangle = \mu \equiv \int\! x p(x)\,dx $$