Does the integral of the probability density function squared mean something?

Question

I've been searching but I couldn't find on the internet if there's any significance for the integral of the pdf squared:

$\int_\mathbb{R} f^2(x)$

That's because, as an alternative of Shannon entropy, you could measure the entropy (or in this case the "order") of a pdf as its distance to the uniform distibution pdf. As it is a "measure" of disimilarity, distance in either way is positive, so, for a discrete probability distribution, its order measure would be defined as:

$\sum_{i}^\mathbb{R} ( p(i) - \frac{1}{\mathbb{R}} )^2 $

A problem with this would be that the order of a discrete distribution would be affected by its range, but if the same formula is considered in a continuous and infinite range, it becomes:

$\int_\mathbb{R} ( f(x) - \frac{1}{\mathbb{R}})^2$

As it is evaluated in an infinite range it becomes:

$\int_\mathbb{R} f^2(x)$

I wanted to know if this would be a valid measure of the entropy (0 for uniform, infinite for something like the dirac delta function) or if there's a flaw I'm not considering for using it in that context.

Answer 1

This is a continuous version of what in the discrete case is called the Simpson index $\sum_i p_i^2$, see Derivation of the Simpson index This is a kind of diversity index, and such an interpretation should be possible in the continuous case also.

Literally, it is the expected value of the probability density function at the observed value of the random variable.