What is language density used for?
The number of words of given length in a language is a very natural parameter. Here are some examples to pique your interest:


The density of $(1+01)^*$ is the Fibonacci sequence.
The density of a regular unary language is eventually periodic.
The density of a regular language is of the form $\Theta(n^k \lambda^n)$ for some integer $k \geq 0$ and real $\lambda \geq 0$.


The parameter $E_L(n)$ measures the entropy per character of a random word of length $n$ in $L$, chosen uniformly at random. For example, if $L$ consists of all binary words containing $\lfloor pn \rfloor$ ones (where $n$ is the length of the words), then $E_L(n)$ converges to $h(p)$, the binary entropy function.