In this paper we consider the notion of generalized density, namely, the natural density of weight $g$ was introduced by Balcerzak et al. (Acta Math. Hungar. {147}(1) (2015) 97-115) and the entire investigation is performed in the setting of probability space extending the recent results of Ghosal (Appl. Math. Comput. { 249} (2014) 502-509) and Das et al. (Filomat {31}(5) (2017) 1463-1473).