Summary: We consider parallel, identical machine scheduling problems, where the jobs are subject to precedence constraints and release dates, and where the processing times of jobs are governed by independent probability distributions. Our objective is to minimize the expected value of the total weighted completion time. Building upon a linear programming relaxation by R. H. Möhring, A. S. Schulz and M. Uetz [J. Assoc. Comput. Mach. 46, 924–942 (1999; Zbl 1176.90262)] and a delayed list scheduling algorithm by C. Chekuri, R. Motwani, B. Natarajan and C. Stein [SIAM J. Comput. 31, 146–166 (2001; Zbl 0992.68066)], we derive the first constant-factor approximation algorithms for this model.