On the least common multiple of random $q$-integers

For every positive integer $n$ and for every $\alpha \in [0,1]$, let $\mathcal{B}(n,\alpha )$ denote the probabilistic model in which a random set $\mathcal{A}\subseteq \{1,\dots ,n\}$ is constructed by picking independently each element of $\{1,\dots ,n\}$ with probability $\alpha$. Cilleruelo, Ru'{e}, \v{S}arka, and Zumalac'{a}rregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $\mathcal{A}$. Let $q$ be an indeterminate and let $[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in $\mathbb{Z}$[q]$bethe$q$-analogofthepositiveinteger$k$. We determine the expected value and the variance of $X := \operatorname{deg} \operatorname{lcm}!\big([$\mathcal{A}$]_q\big)$,where$[$\mathcal{A}$]_q := \big{[k]_q : k \in $\mathcal{A}$\big}$. Then we prove an almost sure asymptotic formula for $X$, which is a $q$-analog of the result of Cilleruelo et al.