The **Fisk distribution**, also called the *log-logistic distribution*, is a continuous probability distribution with many applications: from modeling the distribution of wealth or income in economics, to stream flow rates in hydrology to the lifetime of an organism in biostatistics. It is especially useful in modeling situations where the rate something is happening increases initially, and then after some time begins to decrease.

The term ‘fisk distribution’ is primarily used in economics; in other fields, ‘log-logistic distribution’ is more common.

The fisk distribution has two parameters, a scale parameter and a shape parameter. Both are positive numbers. To say that a random variable x has the log-logistic distribution, we write *x* ~ loglogistic(λ κ). Here λ is the scale parameter and κ is the shape parameter; both, again, are positive.

## The Fisk Distribution’s Probability Density Function

The probability distribution function (pdf) is given by:

where λ is the scale parameter and κ is the shape parameter.

## Other Important Functions

The cumulative distribution function, on the support of *X*, is given by:

The survival function, on the support of *X*, is given by:

The population mean and variance, for a data set that follows the fisk distribution, is given by:

The median of *x* is just 1/λ.

## References

Al-Shomrani, Shawky, Arif & Aslam. Log-logistic distribution for survival data analysis using MCMC. SpringerPlus (2016) 5:1774. DOI 10.1186/s40064-016-3476-7. Retrieved from https://link.springer.com/content/pdf/10.1186/s40064-016-3476-7.pdf on May 18, 2018

Leemis, Larry. Log Logistic Distribution. Retrieved form

http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Loglogistic.pdf on May 18, 2018

World Heritage Encyclopedia, Fisk Distribution. Retrieved from http://self.gutenberg.org/articles/fisk_distribution on May 6, 2018

------------------------------------------------------------------------------**Need help with a homework or test question?** With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. If you'd rather get 1:1 study help, Chegg Tutors offers 30 minutes of **free tutoring** to new users, so you can try them out before committing to a subscription.

If you prefer an **online interactive environment** to learn R and statistics, this *free R Tutorial by Datacamp* is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try *this Statistics with R track*.

**Comments? Need to post a correction?** Please post a comment on our *Facebook page*.

Check out our updated Privacy policy and Cookie Policy