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In life testing and reliability estimation, the underlying failure time distribution need not be homogeneous. It can be a mixture of two distinct lifetime distributions due to two different failure modes. A failure-time distribution is assumed which is the mixture of two exponential distributions. Estimation of the two scale parameters, the mixing parameter, and the reliability is considered. MLEs, and Bayes estimators with respect to proper priors and to Jeffreys' vague priors, are given based on right-censored data. The right-censorship considered here includes types I & I and random right-censoring as special cases. Monte Carlo simulation indicates that the proper Bayes estimators are best with respect to root mean square error (RMSE) under the assumed priors and that the MLEs perform weli overall and are quite satisfactory in practice. The Bayes estimators with respect to Jeffreys' vague priors for the scale parameters do not perform as well as the MLE in the sense of RMSE and tend to have a greater bias.