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What is the most "natural" family of distributions on an interval [0,1] indexed by their mean $\mu$ and standard deviation $\sigma$? I am looking for something occurring in "nature", like normal distributions, except that it has to take values from 0 to 1.

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The reason the normal distribution is so natural is that it is the maximum entropy distribution on the real line with fixed mean and variance. Put another way, in some sense, if you have to assume a distribution and know the mean and variance, it is the minimal assumption. The equivalent idea on an interval is the uniform distribution, which is maximum entropy with no restrictions. I'm not putting this as an answer because I don't actually know the maximum entropy distribution on [0,1] with fixed mean and variance. –  Chris Janjigian Feb 18 at 19:33
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@ChrisJanjigian it seems that the answer is given by Boltzmann's Theorem. Link: en.wikipedia.org/wiki/Maximum_entropy_probability_distribution . It's proportional to $\exp(\lambda_1 x + \lambda_2 x^2)$ for some $\lambda_i$. –  Flounderer Feb 18 at 20:59
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It is almost always more fruitful to approach this question from the other direction: when the phenomenon you are studying is described by a random variable $X$ with support in $[0,1],$ what does the underlying theory suggest about the possible distributions $X$ might have? By inverting this natural question you are stripping all knowledge specific to the phenomenon from your analysis, leaving your results to luck: the luck that the phenomenon just might (accidentally) match some distribution suggested by an abstract mathematical theory of questionable relevance. –  whuber Feb 18 at 21:09
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The problem is that the normal has so many properties that might lead someone to consider it natural for this or that problem that we're left to ponder which properties are most critical.

There's no really 'natural' candidate that's typically indexed by $\mu$ and $\sigma$, though there are two parameter families on the unit interval which have a mean and variance that are functions of the more usual parameters.

 

The Beta distribution family

A very widely used family of two-parameter continuous distributions on the unit interval is the beta family. It should be possible to reparameterize in terms of $\mu$ and $\sigma$, but it would be considerably less 'nice' looking that way.

$$f(x;\alpha,\beta) = \frac{1}{\text{B}(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1};\quad 0\leq x\leq 1,\alpha,\beta>0$$

It has $\mu = \frac{\alpha}{\alpha+\beta}$ and $\sigma^2 = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\,$.

 

Maximum entropy distribution

The maximum entropy distribution with fixed mean and variance on a closed interval appears (via a theorem of Boltzmann's) to be a truncated normal.

Truncated normals are sometimes used in various applications, and are indexed by $\mu$ and $\sigma$*, but I wouldn't say they were usually regarded as the most natural for many problems.

 * but beware! In the truncated normal written in the usual way, the parameters $\mu$ and $\sigma$ aren't the mean and standard deviation of the truncated variable, but of its untruncated parent.

Interestingly, while the beta included the uniform as a special case, the truncated normal includes it as a limiting case.

 

Given the phrasing of your question, those would be the most obvious candidates, and of those, the most widely applied is no doubt the beta family.

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