# Joint probability distributions of variables satisfying a certain constraints

Here is my question:

Given a set of random variables - $\{x_i\}, i=1,2, \dots, n$, and the corresponding pdfs are given by $\{PDF_i\}, i=1,2, \dots, n$.

Now if I were it impose a certain set of constraints on random variables as following

$\{f_1(x_1,x_2, ... x_n)=C_1, f_2(x_1,x_2, ... x_n)=C_2, \dots, f_m(x_1,x_2, ... x_n)=C_m\}$

what will be the joint probability distribution - $PDF(x_1,x_2, ... x_m)=\;?$

Here is an example: The above problem was a generalization of the problems, deriving pdfs in statistical physics, like Maxwell-Boltzmann distribution of ideal gas velocity with a constraint that energy of the system is constant.

• One needs the question to be much more specific before being able to say anything valuable about it. Surely you have a specific setting in mind, let me suggest you explain this setting.
– Did
Commented Nov 23, 2013 at 16:00
• This was a generalization of the problem I am looking for. A simple example would be deriving pdfs like maxwell-blotzmann distribution of velocity by adding a constraint that energy is constant. Commented Nov 25, 2013 at 1:48
• Then "let me suggest you explain this setting" in details in the question.
– Did
Commented Nov 25, 2013 at 7:52

## 1 Answer

Usually there is no way to deduce the joint distribution knowing some constraints except if the constraints are numerous enough to select a single distribution and this will only happen for discrete variables. Usually there are infinitely many solutions.

There is however a theory called "maximum entropy" that might give an answer, that somehow assumes that everything that is not stated by the constraints is ruled by a hidden independence. The principle is : "Take precisely stated prior data (or testable information) about a probability distribution function. Consider the set of all trial probability distributions that would encode the prior data. Of those, the one with maximal information entropy is the proper distribution, according to this principle."