The Simpson index is a variant of diversity index employed in ecology. Given richness, i.e. the number of types, \(R \in \mathbb{N}\) and the proportional abundance \(p_i\) of the \(i\)th type, it is defined by \[\lambda = \sum_{i=1}^R p_i^2.\]
In other words, it is the expectation of the random variable \(X: \Omega \rightarrow \mathbb{R}\) defined by \[X(i) = p_i\] on the probability space \((\Omega, \mathcal{F}, P)\) where \(\Omega = \{1, 2, ..., R\}, \mathcal{F}\) is the power set of \(\Omega\), and \(P(S) = \sum_{i \in S} p_i.\)
It follows that we can extend the Simpson index to the case with a continuous set of types \(\Omega\) in a straightforward manner; that is, given \(X: \Omega \rightarrow \mathbb{R}\) with probability density function \(f\), \[\lambda = \mathbb{E}[f(X)] = \int_\Omega f \circ X dP = \int_\mathbb{R} \lvert f(x) \rvert^2 dx,\] which is just the squared \(L^2\) norm of \(f\), granted that \(f\) is square-integrable.