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The probability density function of the arrival time of \v{C}erenkov light ona photo-multiplier tube has been studied. This study covers light production,transmission and detection. The light production includes the light from amuon, the light from a shower and the light due to the energy loss of a muon.For the transmission of light, the effects of dispersion, absorption andscattering in the medium are considered. For the detection of light, theangular acceptance and the quantum efficiency of the photo-multiplier tube aretaken into account.