In this experiment, a cosmic ray muon detector is designed based on plastic scintillator and photomultiplier with fast time response. The waveform of electrical pulse signal produced by charged particle in the detector is read out and recorded in computer by a digitizer. Through offline data analysis, the procedures of muon decay event selection, timing of pulse signal and lifetime fitting are completed. In this experiment, the rise time of pulse signal is less than 20 ns, and the noise is less than 3 mV. It is estimated that the time resolution of detector can reach an order of nano second by using leading-edge timing method. The measured average lifetime of the muon is 2.188±0.045 μs, which is consistent with the world average of muon lifetime with respect to the experimental error.