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We study some properties of binary sequences generated by randomsubstitutions of constant length. Specifically, assuming the alphabet$\{0,1\}$, we consider the following asymmetric substitution rule of length$k$: $0 \to \langle 0, 0, \ldots,0\rangle$ and $1 \to \langle Y_1, Y_2, \ldots,Y_k \rangle$, where $Y_i$ is a Bernoulli random variable with parameter $p \in[0,1]$. We obtain by recurrence the discrete probability distribution of thestochastic variable that counts the number of ones in the sequence formed aftera number $i$ of substitutions (iterations). We derive its first two statisticalmoments, mean and variance, and the entropy of the generated sequences as afunction of the substitution length $k$ for any successive iteration $i$, andcharacterize the values of $p$ where the maxima of these measures occur.Finally, we obtain the parametric curves entropy-variance for each iterationand substitution length. We find two regimes of dependence between these twovariables that, to our knowledge, have not been previously described. Besides,it allows to compare sequences with the same entropy but different variance andvice versa.

Entropy-Variance curves of binary sequences generated by random substitutions of constant length