The log-logistic distribution is a continuous probability distribution used in statistics and reliability analysis. It is particularly useful for modeling the distribution of positive random variables, especially in contexts where the data exhibits a skewed distribution and has a long right tail. The log-logistic distribution is often employed in survival analysis and economics. ### Definition: A random variable \(X\) follows a log-logistic distribution if its logarithm, \(\log(X)\), follows a logistic distribution.
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For a scale and shape , the log-logistic distribution has cumulative distribution function and probability density functionIt has no point mass at zero. Its quantile function is for , so its median is . The log odds relation explains both its name and the log-logistic probability plot.[ref]
SciPy User Guide: Fisk (Log Logistic) Distribution. CDF, density and inverse CDF, with the scale factor restored