Difference between standard beta and unstandard beta distributions?

Question

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?

Answer 1

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.