In this paper, we investigate the resulting probability distribution of a kriging model when the values of the model parameters must be estimated from observations of the process being modeled. The output of a kriging model defines a Gaussian probability distribution when the values of the model parameters are given. In practice, these model parameters must be estimated from observations of the process being modeled (typically a computationally expensive computer model). We found that when the model parameters are treated as random variables instead of known values, the resulting probability distribution of the kriging model can be well approximated by a Student-t distribution, a distribution with fatter tails than the Gaussian or normal distribution. The Markov chain Monte Carlo (MCMC) method was used to determine the probability distributions of the model parameters and the output of the kriging model given the observations. The resulting model parameters were validated against the results of a Bayesian analysis of a simple one-dimensional test problem. The results were also compared to the standard method of Maximum Likelihood Estimation as an alternative method to estimate model parameters.