The IceCube collaboration recently reported the potential detection of two cascade neutrino events in the energy range 1-10 PeV .
We study the possibility that these PeV neutrinos are produced by gamma-ray bursts ( GRBs ) , paying special attention to the contribution by untriggered GRBs that elude detection due to their low photon flux .
Based on the luminosity function , rate distribution with redshift and spectral properties of GRBs , we generate , using Monte-Carlo simulation , a GRB sample that reproduce the observed fluence distribution of Fermi/GBM GRBs and an accompanying sample of untriggered GRBs simultaneously .
The neutrino flux of every individual GRBs is calculated in the standard internal shock scenario , so that the accumulative flux of the whole samples can be obtained .
We find that the neutrino flux in PeV energies produced by untriggered GRBs is about 2 times higher than that produced by the triggered ones .
Considering the existing IceCube limit on the neutrino flux of triggered GRBs , we find that the total flux of triggered and untriggered GRBs can reach at most a level of \sim 10 ^ { -9 } GeVcm ^ { -2 } s ^ { -1 } sr ^ { -1 } , which is insufficient to account for the reported two PeV neutrinos .
Possible contributions to diffuse neutrinos by low-luminosity GRBs and the earliest population of GRBs are also discussed .