A Monte Carlo investigation of Markov chain matrices was conducted to create empirical distributions for two statistics created from the transition matrices. Curve fitting techniques developed by Karl Pearson were used to deduce if theoretical equations could be fit to the two sets of distributions. The set of distributions which describe the variance among the transition probabilities in the steady state condition was discovered to be a member of the Beta distribution family. The set of distributions for t, the statistic which describes the number of transitions required to reach the steady state condition, could not be fit to any known distribution function using the Pearson techniques. Empirical results agreed very closely with the theoretical predictions. (Three types of Markov chains--regular, absorbing, and cyclic--are discussed within the paper. Information relative to the percentage of occurence of each type of chain within the population of transition matrices is presented, along with a comparison of the occurence of each type within the samples of 5,000 and 20,000 matrices generated). (Author/CTM)