Theoretical derivation and numerical calculation achieve the quantum dot numerically exact multi-photon emission spectra starting from the interaction of quantum dot and cavity,using the master equation that includes incoherent pumping and pure dephasing and combines with the quantum regression theorem, and considering the relevant initial conditions.Then comparing the multi-photon(fermion statistics) spectra with the one-thoton(boson system) correlation level, our analysis shows that according to the fermion statistics the pure dephasing, and the thermal bath model, an excellent fitting to the data is obtained with the recent experimental of Ref[22] results in a quantum dot-micropiller cavity system. The theory and the recent experiments both show pump-induced stimulated emission and anharmonic cavity-QED in quantum dot-cavity systems.