The paper by Mark Bagnoli and Ted Bergstrom explores the properties and applications of log-concave probability distributions. Log-concave functions, where the natural logarithm is concave, have significant implications in various fields such as economics, political science, biology, and industrial engineering. The authors catalog and prove a series of theorems relating log-concavity and log-convexity of probability density functions, distribution functions, reliability functions, and their integrals. They examine the invariance of these properties under integration, truncations, and other transformations, and relate them to reliability functions, failure rates, and the mean residual lifetime function. The paper also introduces the "mean-advantage-over-inferiors function" for truncated distributions and discusses its monotonicity in relation to log-concavity or log-convexity. Additionally, the authors review a wide range of commonly used probability distributions and their log-concavity or log-convexity properties. The paper aims to provide a unified exposition of related results and sample applications of this theory, building on earlier work in statistics, economics, and industrial engineering.The paper by Mark Bagnoli and Ted Bergstrom explores the properties and applications of log-concave probability distributions. Log-concave functions, where the natural logarithm is concave, have significant implications in various fields such as economics, political science, biology, and industrial engineering. The authors catalog and prove a series of theorems relating log-concavity and log-convexity of probability density functions, distribution functions, reliability functions, and their integrals. They examine the invariance of these properties under integration, truncations, and other transformations, and relate them to reliability functions, failure rates, and the mean residual lifetime function. The paper also introduces the "mean-advantage-over-inferiors function" for truncated distributions and discusses its monotonicity in relation to log-concavity or log-convexity. Additionally, the authors review a wide range of commonly used probability distributions and their log-concavity or log-convexity properties. The paper aims to provide a unified exposition of related results and sample applications of this theory, building on earlier work in statistics, economics, and industrial engineering.