We discuss properties of distributions that are multivariate totally positive of order two (MTP2). Such distributions appear in the context of positive dependence, ferromagnetism in the Ising model, and various latent models. We show that maximum likelihood estimation for MPT2 exponential families is a convex problem. Hence, if the MLE exists, it is unique. In the Gaussian setting we prove that the MLE exists with only 2 observations and that MTP2 implies sparsity of the concentration matrix without the need of a tuning parameter. This makes MTP2 interesting in the high-dimensional setting, in particular as an alternative to the graphical lasso.