I Separation of variables for Named Probability Density Distributions

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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}
 
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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'?
 
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) ##?
 
pbuk said:
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
 
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