Minimizer schemes, or just minimizers, are a very important computational primitive in sampling and sketching biological strings. Assuming a fixed alphabet of size \(\sigma \) , a minimizer is defined by two integers \(k,w\ge 2\) and a total order \(\rho \) on strings of length k (also called k-mers). A string is processed by a sliding window algorithm that chooses, in each window of length \(w+k-1\) , its minimal k-mer with respect to \(\rho \) . A key characteristic of the minimizer is the expected density of chosen k-mers among all k-mers in a random infinite \(\sigma \) -ary string. Random minimizers, in which the order \(\rho \) is chosen uniformly at random, are often used in applications. However, little is known about their expected density \(\mathcal{D}\mathcal{R}_\sigma (k,w)\) besides the fact that it is close to \(\frac{2}{w+1}\) unless \(w\gg k\) .    We first show that \(\mathcal{D}\mathcal{R}_\sigma (k,w)\) can be computed in \(O(k\sigma ^{k+w})\) time. Then we attend to the case \(w\le k\) and present a formula that allows one to compute \(\mathcal{D}\mathcal{R}_\sigma (k,w)\) in just \(O(w\log w)\) time. Further, we describe the behaviour of \(\mathcal{D}\mathcal{R}_\sigma (k,w)\) in this case, establishing the connection between \(\mathcal{D}\mathcal{R}_\sigma (k,w)\) , \(\mathcal{D}\mathcal{R}_\sigma (k+1,w)\) , and \(\mathcal{D}\mathcal{R}_\sigma (k,w+1)\) . In particular, we show that \(\mathcal{D}\mathcal{R}_\sigma (k,w)<\frac{2}{w+1}\) (by a tiny margin) unless w is small. We conclude with some partial results and conjectures for the case \(w>k\) .