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 \( \mu \) $$ \begin{equation*} \langle x\rangle = \mu \equiv \int x p(x)dx, \end{equation*} $$ for a continuous distribution and $$ \begin{equation*} \langle x\rangle = \mu \equiv \sum_{i=1}^N x_i p(x_i), \end{equation*} $$ for a discrete distribution. Qualitatively it represents the centroid or the average value of the PDF and is therefore simply called the expectation value of \( p(x) \).