# Common Continuous Distributions with [0,1] support

## Question

I am looking to understand what possible common statistical continuous distributions exist with support [0,1].

## Background

In my work I often come across data which are bounded between 0 and 1 (both inclusive) and likely skewed to the right.

This data mainly consist of sales converted into percentages between 0 and 1, by either calculating total per cent of sales or conversion (sales divided by page views).

As I am not very proficient in statistics, I always struggle to find the best distribution to explain this data.

• Could you be more specific? What exactly are you doing? What is your data? Notice that you can re-scale and/or truncate almost any distribution so that it is bounded in [0,1], moreover, you can use zero-inflated mixtures etc. – Tim Mar 27 '17 at 10:10

Wikipedia has a list of distributions supported on an interval

Leaving aside mixtures and 0-inflated and 0-1 inflated cases (though you should definitely be aware of all of those if you model data on the unit interval), which ones are common would be hard to establish (it will vary across application areas for example), but the beta family, and the triangular, and the truncated normal would probably be the main candidates as they seem to be used in a variety of situations.

Each of them can be defined on (0,1) and can be skewed either direction.

One example of each is shown here:

That they're often used doesn't imply they'll be suitable for whatever situation you're in, though. Model choice should be based on a number of considerations, but where possible, theoretical understanding and practical subject area knowledge are both important.

I always struggle to find the best distribution to explain this data.

You should get away from worrying about "best", and focus on "sufficient/adequate for the present purpose". No simple distribution such as the ones I mentioned will really be a perfect description of real data ("all models are wrong..."), and what might be fine for one purpose ("... some are useful") may be inadequate for some other purpose.

If you have exact zeros (or exact ones, or both), then you will need to model the probability of those 0's and use a mixture distribution (a 0-inflated distribution if you can have exact 0's) -- shouldn't use a continuous distribution.

It's not really all that hard to deal with simple mixtures. You'll no longer have a density but the cdf is not much more effort to write down or evaluate than it would be in the continuous case; similarly quantiles are not much more effort either; means and variances are almost as readily calculated as before; and they're easy to simulate from.

Taking an existing continuous distribution on the unit interval and adding a proportion of zeros (and/or ones) is on the whole a pretty convenient way to model proportions that are mostly continuous but can be 0 or 1.

• Thank you. I did look at the wikipedia, but didn't really know where to go from there, particularly, as I tried the Beta in R and it seemed to not like my 0's as errors occurred. So I thought I might have read the Wiki page wrong. I will have a look at the 3 you mentioned (incl. going back to Beta), and I do understand they might not be suitable - it is just better to start with something and then see if "sufficient/adequate for the present purpose". :) – Alex Mar 27 '17 at 10:45
• I think Tim just answered why I struggled with the Beta in R. :) – Alex Mar 27 '17 at 10:47
• This seems fine, but I think the OP is confused about the nature of his data. He asks about "continuous" distributions (which is therefore what you discuss), but says his data are "sales converted into percentages between 0 and 1, by either calculating total per cent of sales or conversion (sales divided by page views)", that suggests the distributions aren't really continuous number of sales / number of page views is probably binomial or some variant thereof. – gung - Reinstate Monica Mar 27 '17 at 14:43
• @gung you should convert this comment into answer. – Tim Mar 27 '17 at 16:27
• @gung yes, OP changed the question after I answered it. I did make some changes after that. – Glen_b Mar 28 '17 at 0:45

Adding to Glen_b's answer, notice that if you are dealing with a continuous random variable, then in theory it shouldn't really matter if the distribution supports $[0, 1]$, or $(0, 1)$ bounds as $\Pr(X=0) = \Pr(X=1) = 0$ (see $P[X=x]=0$ when $X$ is continuous variable). In real life you meet exact zeros and ones due to measurement precision issues and the common workaround is to apply the simple "squeezing" transformations to move them away from the bounds (see Dealing with 0,1 values in a beta regression and Beta regression of proportion data including 1 and 0). See also then Why exactly can't beta regression deal with 0s and 1s in the response variable? thread for related discussion.

So inclusive bounds should not concern you that much when considering common bounded distributions like beta, Kumarshwamy, triangular distribution etc.

If, as you are saying, your data has exact zeros for other reasons then measurement precision issues, then you are dealing with mixed-type data and you should consider zero-inflated models, i.e. using mixture distribution in form

$$g(x) = \begin{cases} \pi + (1-\pi) f(x) & x = 0 \\ (1-\pi) f(x) & x > 0 \end{cases}$$

where $f$ is non-zero-inflated distribution and $\pi$ is the mixing parameter controlling for the probability of excess zeros in your data, what follows is that if $f(0)=0$, then $g(0) = \pi$ for distributions $f$ with non-inclusive bounds. You can easily extend this line of reasoning to zero-and-one inflated model etc.

• I think you just answered one of my main problems, when I stared looking at this on wiki. Thank you. I had already tried the Beta in R and I couldn't understand why it didn't like 0's. I actually just thought it meant that it supports (0,1), instead of [0,1].. silly me.. :) Thank you. – Alex Mar 27 '17 at 10:49
• As my data genuinely has real 0's (not due to rounding), am I incorrect in using only a continuous distribution and not a mixture distribution? I prefer your squeeze solutions, mainly because I really am not proficient enough in stats to work with mixture distributions. Thanks again. – Alex Mar 27 '17 at 11:00
• Ye s, if you have exact 0's (or exact 1's, or both) you shouldn't use a continuous distribution. When you take a mixture by having a proportion of 0s and a continuous distribution otherwise, it's often called a zero-inflated distribution (or model). It's not really all that hard to deal with simple mixtures like that. You no longer have a density but the cdf is not very much more effort to write down that it would be in the continuous case, means and variances are almost as readily calculated as before and so on. If you have specific questions about mixtures, you have a place to ask about them – Glen_b Mar 27 '17 at 11:12