Separation of variables for Named Probability Density Distributions

[redtree]

TL;DR

What are the named Probability Density Distributions for which separation of variables can be performed?

Given a probability density distribution $P(\vec{x})$, for what named distributions is the following true:
\begin{equation}
\begin{split}
P(\vec{x}) &= P_1(x_1) P_2(x_2) ... P_n(x_n)
\end{split}
\end{equation}

 

[pbuk]

Do you think that that equality is determined by the distribution or by some property of $\vec{x}$?

Do you have a complete list of 'named distributions'?

 

[redtree]

Yes; it's a bit of a list. From the following: http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf

My list includes any distribution where the observable is limited to the exponential term.

 

[pbuk]

So what condition is necessary and sufficient for $P(x_1, x_2, ... ,x_n) = P_1(x_1) P_2(x_2) ... P_n(x_n)$?

 

[BWV]

So what condition is necessary and sufficient for $P(x_1, x_2, ... ,x_n) = P_1(x_1) P_2(x_2) ... P_n(x_n)$?

Independence