Centrality measures play an important role in determining the importance of nodes in networks. For strongly connected networks, the random walk centrality measures how easy it is to reach a given state from another randomly chosen state. This measure requires calculating a generalized group inverse for the transition matrix, which can be computationally difficult for large state spaces. It is known that the random walk centrality for a particular state can be written as a function of the first and second moments of the first passage times for that state. In this study, using realization of random walks, we estimate the distributions of first passage times by using a number of statistical methods, including Bayesian bootstrap and two Poisson mixture model approaches. Finally, we compare the resulting estimates of the Random walk centrality measures to the true values.