For every positive integer n and for every α ∈ [ 0 , 1 ] documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$lpha in [0, 1]$$end{document} , let B ( n , α ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathcal {B}}(n, lpha )$$end{document} denote the probabilistic model in which a random set A ? { 1 , … , n } documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathcal {A}} subseteq {1, ldots , n}$$end{document} is constructed by picking independently each element of { 1 , … , n } documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${1, ldots , n}$$end{document} with probability α documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$lpha $$end{document} . Cilleruelo, Rué, ?arka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of A documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathcal {A}}$$end{document} .Let q be an indeterminate and let [ k ] q : = 1 + q + q 2 + ? + q k - 1 ∈ Z [ q ] documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$[k]_q := 1 + q + q^2 + cdots + q^{k-1} in {mathbb {Z}}[q]$$end{document} be the q -analog of the positive integer k . We determine the expected value and the variance of X : = deg lcm ( [ A ] q ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$X := deg {ext {lcm}}!ig ([{mathcal {A}}]_qig )$$end{document} , where [ A ] q : = { [ k ] q : k ∈ A } documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$[{mathcal {A}}]_q := ig {[k]_q : k in {mathcal {A}}ig }$$end{document} . Then we prove an almost sure asymptotic formula for X , which is a q -analog of the result of Cilleruelo?et?al.
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