Abstract: A stochastic model of detective problem was built and was transformed into the problems of finding the distributions and the numerical characters of inquiry number and the problems of finding the limit distributions of the Markov chain．As to the cases in which the criminal makes a confession of his crime‚the probability distribution of inquiry number for cracking a criminal case was calculated and the formula to calculate the mean time and the variance of inquiry number was given‚from which a new identity resulted．As to the cases in which the criminal denies his crime‚the one-step transform probability matrix in Markov chain and the final distribution of inquiry number for cracking a criminal case were obtained．The same results were obtained by the methods of characteristic functions． 

Key words: detective model, detective identity, markov chain, limit distribution