We consider a multivariate survival distribution derived from an inverse Gaussian mixture of exponential distributions. The variables of this multivariate distribution are shown to exhibit total positive dependence of order 2. A general formula for joint moments and the monotonicity properties of hazard rates are described. Inference methods—including derivation of a posterior distribution for the unknown exponential hazard rate, maximum likelihood estimation of the mixture distribution parameters, and derivation of a posterior predictive distribution for a new observation—are developed. Procedures for assessing model adequacy are also presented. The computational requirements of the methods are modest, and censored data are handled with ease. The inference methods and model assessment procedures are illustrated with several case examples, one of which exploits the connection between this multivariate distribution and the inverse Gaussian mixture of Poisson distributions.