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

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.

• You might want to have a look at the Gini entropy that is a special case of the Tsallis entropy Commented Nov 19, 2017 at 3:27
• Thanks! From what I've read so far, the continuous formula for Gini entropy fits the same criteria. Commented Nov 19, 2017 at 20:18

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.