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What is the difference between standard beta and unstandard beta distributions? How to understand in an article if it is not described if it is standard or not?

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Standard beta distribution is beta distribution bounded in $(0, 1)$ interval, so it is what we generally refer to when talking about beta distribution. Beta is not standard if it has other bounds, denoted sometimes as $a$ and $b$ (lower and upper bound), you can find some information here.

So the general form of probability density function is

$$ f(x) = \frac{(x-a)^{\alpha-1}(b-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta) (b-a)^{\alpha+\beta-1}} $$

while in most cases we refer to standard beta, i.e.

$$ f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}} { \mathrm{B}(\alpha,\beta)} $$

If $X$ is beta distributed with bounds $a$ and $b$, then you can transform it to standard beta distributed variable $Z$ by simple normalization

$$ Z = \frac{X-a}{b-a} $$

It is also easy to back-transform standard beta to beta with $a$ and $b$ bounds by

$$ X = Z \times (b-a) + a $$

So to compute pdf, cdf, or random number generation for non-standard beta, you need only the basic functions and formulas for beta distribution. If you want to use density function of standard beta with non-standard beta just remember to normalize the density, i.e. $f(\frac{X-a}{b-a})/(b-a)$.

In most cases people referring to beta distribution are talking about standard beta distribution. If the distribution has different bounds than $(0, 1)$, than it is obviously not a standard beta, so it should be clear from context.

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  • $\begingroup$ Thanks for this. Can I convert the ust Beta to Std beta by using Standard error? $\endgroup$ – hero1985 Dec 12 '15 at 23:03
  • $\begingroup$ @hero1985 using SE no, but if you can using bounds, see my edit. $\endgroup$ – Tim Dec 13 '15 at 7:57
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    $\begingroup$ Readers note that the section of the wikipedia beta distribution article relating to the Four parameter beta distribution may also be helpful. $\endgroup$ – Glen_b Jul 28 '16 at 11:34
  • $\begingroup$ Maybe it is obvious to people more mathematically inclined than me, but what does the B in the first two equations stand for? Cheers. EDIT: opened the "here" link and found it. nvm $\endgroup$ – bunsenbaer Aug 16 '18 at 19:28
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    $\begingroup$ @bunsenbaer beta function en.wikipedia.org/wiki/Beta_function $\endgroup$ – Tim Aug 16 '18 at 19:34

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