Integral of Joint Beta Distribution I have a joint pdf  of three independent beta RVs $\{\theta_1, \theta_2, \theta_3\}$ with separate parameters $\{(\alpha_1,\beta_1), (\alpha_2,\beta_2), (\alpha_3,\beta_3)\}$
I need to find pdf $g(\theta_1, \theta_2, \theta_3)$ such that $\theta_1 \le \theta_2 \le \theta_3$.
Am I correct in expressing $g(\ \ )$ as
$g= f(\theta_1, \theta_2, \theta_3|\ \theta_1 \le \theta_2 \le \theta_3) = \frac{f(\theta_1, \theta_2, \theta_3)}{P(\theta_1 \le \theta_2 \le \theta_3)}$
$P(\theta_1 \le \theta_2 \le \theta_3) = \int_{{\theta_3}=0}^1 \int_{{\theta_2}=0}^{\theta_3}\int_{{\theta_1}=0}^{\theta_2} f(\theta_1, \theta_2, \theta_3)\ \ d{\theta_1}d{\theta_2}d{\theta_3}$
Is there any closed for solution to this (assuming that Im correct in the expressions)
If not can I simulate the solution?
 A: Have you considered studying multi-variate beta distribution first?
https://journals.sagepub.com/doi/abs/10.3102/10769986007004271?journalCode=jeba
Once you have, you should be able to formulate your cumulative beta distribution, under the constraints you're interested in, and from there (through differentiation), your  PDF.
As far as a simulation-based answer is concerned, you should definitely be able to do this if an implementation of the joint distribution existed. If no such distribution has been implemented, how would you be able to sample it in a stochastic simulation?
There's also another problem lurking in your question. I don't have access to the paper above but I would imagine there are some correlation constants besides the alphas and betas for the joint, 3-dimensional beta distribution.
The construction of bi-variate Beta distribution has been discussed here:
https://arxiv.org/pdf/1406.5881.pdf
This second paper also discusses extensions to larger number of dimensions (3 and more).
This second paper is obviously more easily obtainable.
This paper gives a review of the previous work. For example, one sees that the number of parameters for the join distribution is not necessarily 2K, where K is the number of random variables.
