The Muth distribution is a continuous random variable introduced in the context of reliability theory. In this paper, some mathematical properties of the model are derived, including analytical expressions for the moment generating function, moments, mode, quantile function and moments of the order statistics. In this regard, the generalized integro-exponential function, the Lambert W function and the golden ratio arise in a natural way. The parameter estimation of the model is performed by the methods of maximum likelihood, least squares, weighted least squares and moments, which are compared via a Monte Carlo simulation study. A natural extension of the model is considered as well as an application to a real data set.