In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If X is a random variable with a Cauchy distribution, then Y = exp(X) has a log-Cauchy distribution; likewise, if Y has a log-Cauchy distribution, then X = log(Y) has a Cauchy distribution. Characterization The log-Cauchy distribution is a special case of the log-t distribution where the degrees of freedom parameter is equal to 1. Probability density function The log-Cauchy distribution has the probability density function: where is a real number and . If is known, the scale parameter is . and correspond to the location parameter and scale parameter of the associated Cauchy distribution. Some authors define and as the location and scale parameters, respectively, of the log-Cauchy distribution. For and , corresponding to a standard Cauchy distribution, the probability density function reduces to: Cumulative distribution function The cumulative distribution function (cdf) when and is: Survival function The survival function when and is: Hazard rate The hazard rate when and is: The hazard rate decreases at the beginning and at the end of the distribution, but there may be an interval over which the hazard rate increases. Properties The log-Cauchy distribution is an example of a heavy-tailed distribution. Some authors regard it as a "super-heavy tailed" distribution, because it has a heavier tail than a Pareto distribution-type heavy tail, i.e., it has a logarithmically decaying tail. As with the Cauchy distribution, none of the non-trivial moments of the log-Cauchy distribution are finite. The mean is a moment so the log-Cauchy distribution does not have a defined mean or standard deviation. The log-Cauchy distribution is infinitely divisible for some parameters but not for others. Like the lognormal distribution, log-t or log-Student distribution and Weibull distribution, the log-Cauchy distribution is a special case of the generalized beta distribution of the second kind. The log-Cauchy is actually a special case of the log-t distribution, similar to the Cauchy distribution being a special case of the Student's t distribution with 1 degree of freedom. Since the Cauchy distribution is a stable distribution, the log-Cauchy distribution is a logstable distribution. Logstable distributions have poles at x=0. Estimating parameters The median of the natural logarithms of a sample is a robust estimator of . The median absolute deviation of the natural logarithms of a sample is a robust estimator of .