The Weibull distribution if the shape parameter is >= 1. Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave. The following distributions are non-log-concave for all parameters: The Student's t-distribution. The Cauchy distribution. The Pareto distribution. The log-normal distribution. The F-distribution. Note that the cumulative distribution function (CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's: The log-normal distribution. The Pareto distribution. The Weibull distribution when the shape parameter < 1. The gamma distribution when the shape parameter < 1. The following are among the properties of log-concave distributions: If a density is log-concave, so is its cumulative distribution function (CDF). If a multivariate density is log-concave, so is the marginal density over any subset of variables. The sum of two independent log-concave random variables is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave. The product of two log-concave functions is log-concave. This means that joint densities formed by multiplying two probability densities (e.g. the normal-gamma distribution, which always has a shape parameter >= 1) will be log-concave. This property is heavily used in general-purpose Gibbs sampling programs such as BUGS and JAGS, which are thereby able to use adaptive rejection sampling over a wide variety of conditional distributions derived from the product of other distributions. If a density is log-concave, so is its survival function. If a density is log-concave, it has a monotone hazard rate (MHR), and is a regular distribution since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e. which is decreasing as it is the derivative of a concave function. See also logarithmically concave sequence logarithmically concave measure logarithmically convex function convex function Notes References Mathematical analysis Convex analysis