We present an integration based procedure for predicting the distribution f of an indicator of interest in situations where, in addition to the sample data, one has access to covariates that are available for the entire population. The proposed method, based on similar ideas that have been used in the literature on policy evaluation, provides an alternative to existing simulation and imputation methods. It is very simple to apply, flexible, requires no additional assumptions, and does not involve the inclusion of artificial random terms. It therefore yields reproducible estimates and allows for valid inference. It also provides a tool for future predictions, scenarios and ex-ante impact evaluation. We illustrate our procedure by predicting income distributions in a case with sample selection, and both current and future doctor visits. We find our approach outperforms other commonly used procedures substantially.