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In this paper we study, in the context of weighted function spaces of continuous functions, a class of integral operators of probabilistic type, which are constructed by means of the Weibull distribution, and a suitable modification that preserves linear functions. We study the approximation properties of those two sequences of operators and we present estimates of the rate of convergence, as well as some asymptotic formulae. In particular, we prove that the proposed modification has a better rate of convergence. Finally, we show how the modified Weibull operators can be fruitfully employed to approximate suitable positive C 0 -semigroups. [ABSTRACT FROM AUTHOR]