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The W and Z boson production was measured via the muonic decay channel in proton-lead collisions at sNN−−−√=5.02 TeV at the Large Hadron Collider with the ALICE detector. The measurement covers backward (−4.46<ycms<−2.96) and forward (2.03<ycms<3.53) rapidity regions, corresponding to Pb-going and p-going directions, respectively. The Z-boson production cross section, with dimuon invariant mass of 60<mμμ<120 GeV/c2 and muon transverse momentum (pμT) larger than 20 GeV/c, is measured. The production cross section and charge asymmetry of muons from W-boson decays with pμT>10 GeV/c are determined. The results are compared to theoretical calculations both with and without including the nuclear modification of the parton distribution functions. The W-boson production is also studied as a function of the collision centrality: the cross section of muons from W-boson decays is found to scale with the average number of binary nucleon-nucleon collisions within uncertainties.