This paper presents an optimization model and solution procedure for planning investments in gas distribution networks for residential customers. The situation can be considered a capital budgeting problem under uncertainty. There is uncertainty about whether a potential customer will convert to gas service if a distribution main is built, the revenue generated if the household does convert, and the cost of constructing the main. A fixed annual budget is allocated to a set of discrete, competing projects over time. The allocation is done by maximizing the expected net present value (NPV) given the decision-maker's risk preferences. The probability distribution of the NPV for each competing project is created from two statistical models. A binary probit model is used to estimate the probability of conversion for a potential customer. A random effects regression model is used to estimate the revenue generated should a particular potential customer switch to gas. A rollout value greedy heuristic was devised to solve the resulting optimization formulation. Two case studies based on data from a large gas company illustrate the analysis.