The research presented in this dissertation focuses on the measurement and analysis of subjective probability distributions over stochastic outcomes, a central issue in the study of decision-making under uncertainty. The empirical setting is rural Tanzania, where the degrees of risk and uncertainty characterizing both human capital and productive investment decisions are exacerbated by widespread dependence on rain-fed agriculture, inadequate social safety nets, and a poorly developed information infrastructure. I present a sequence of methodological, theoretical and empirical chapters in which I estimate subjective returns distributions in an existing data set, develop and explain a new method of collecting subjective distributions data, characterize the information content of the data collected, and make use of the data to estimate a structural agricultural production model. Chapter 1 explores the role of estimated, rather than measured, subjective returns to education in schooling choice decisions. Using an existing panel survey from Tanzania, I estimate earnings-education distributions separately for 1991, 2004 and 2010. I then use individual-level predictions of the first two moments of the earnings distribution to estimate a random effects probit model on binary enrollment decisions for school-aged children in the years 1991-1994. I find that the returns to education have been and remain high for women, while for men the returns increased over the twenty study years to nearly match those of women. In addition, the probability of enrollment is increasing in the subjective conditional expectation of earnings, and decreasing in the subjective conditional variance of earnings. Chapter 2 describes the phone-based survey method that I used to gather subjective probability distributions data from a sample of Tanzanian cotton farmers. I describe the various technical issues faced in the implementation of this method, outline the lessons learned and the numerous refinements made over the course of the study, and speculate on the feasibility of phone-based data collection in other settings in low income countries. In Chapter 3, I analyze the information content of subjective distributions data gathered in the way that has become standard in development economics, i.e., by having respondents allocate a fixed number of counters to boxes that represent the intervals of a histogram. I use inference about the respondents' choice problem to analyze the partial identification of the underlying belief. I provide bounds on the density in subsets of intervals, provide bounds on the underlying CDF, define the joint identification region for the measure vector, and develop and implement a feasible numerical method for jointly bounding the moments of the unobserved distribution. I also provide simulation evidence for the optimal design of survey instruments and the optimal way to approximate these data with smooth distributions. Lastly, Chapter 4 makes use of the regularly spaced within-season measures of subjective yield and price distributions collected from Tanzanian cotton farmers to study the farmer's dynamic resource allocation problem. Using these data, I develop a novel method for estimating a stochastic production function when error parameters are observed at the plot-level throughout the cultivation season.