By virtue of kinetic proofreading theory, the puzzles of selectivity and specificity common in cellar activities are successfully explained. The chemical master equation is adopted to describe stochastic processes of kinetic proofreading models with two and three middle steps. Stochastic properties around the steady state are analyzed with the linear noise approximation under various parameters. Relations between noise density and parameters are revealed with numerical simulation. The result suggest that noises propagate and escalate as reactions proceed, furthermore multiple reaction rates determine noise intensity.