We consider positive linear operators of probabilistic type L,f acting on real functions / defined on the positive semi-axis. We deal with the problem of uniform convergence of L,f to / , both in the usual sup-norm and in a uniform L p type of norm. In both cases, we obtain direct and converse inequalities in terms of a suitable weighted first modulus of smoothness of/. These results are applied to the Baskakov operator and to a gamma operator connected with real Laplace transforms, Poisson mixtures and Weyl fractional derivatives of Laplace transforms.1991 Mathematics subject classification (Amer. Math. Soc): primary 44A10,41A35, 60J30.