# Distributions bounded on both sides

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Common name for distributions that are bounded on one side

I am looking for common name or an exhaustive list of which distributions are bounded on both sides [0,1]. This is because my data is similar to this (cannot be less than 0 and greater than 100). I need to fit a distribution to this data. In order to do this, I need to firs find a list of distributions that are bounded on both sides and try them one by one to see which distribution closely resembles my data.

• There is no exhaustive list. Think of this: any nonnegative function with finite integral on [0,1] can be a density function. You can't name them all. – Aksakal Mar 14 '18 at 14:19
• Okay. Is there a list of some commonly used family of distribution that I can fit to a bounded dataset – 89_Simple Mar 14 '18 at 14:26
• Have a look at beta regression and if you analyze your data in R, the betareg package is very useful. Also searching for beta regression on this site will give you a lot of examples. Note however, that if your data has zero and ones, the analysis won't run. In this case, you can scale your data so it will fall between 0 and 1. There is a scaling procedure mentioned in the PDF (page 3, 3rd paragraph). – Stefan Mar 14 '18 at 15:42

## 2 Answers

There's no one universal list, and there can't be an exhaustive list. However, you can find a list of some continuous densities with bounds in wiki: https://en.wikipedia.org/wiki/List_of_probability_distributions#Supported_on_a_bounded_interval

Also, remember that you can take any distribution and bound it in an interval. These are called truncated distributions, e.g. see truncated normal.

• Thank you. I was not aware of the truncated distribution. I will read around it. – 89_Simple Mar 14 '18 at 14:32

Check out the kernel functions. When a kernel satisfies $$\int_{-\infty}^\infty K(x)\,\mathrm dx = 1,$$ it results in a probability density function.