A Bayesian approach to the estimation of the parameters of a two-parameter exponential distribution and the reliability function associated with it is developed using censored samples. Bayesian point estimates are obtained for these three quantities and it is shown that under a suitable choice of the prior distribution and of the loss function they are approximately equivalent to the corresponding maximum likelihood estimates (MLE) and minimum variance unbiased estimates (MVUE). Also, Bayesian probability points and confidence points are obtained for both the parameters and their approximate equivalence is brought out.