I Separation of variables for Named Probability Density Distributions

AI Thread Summary
The discussion focuses on the conditions under which a joint probability density distribution can be expressed as the product of individual distributions. It questions whether this factorization is determined by the properties of the distributions themselves or by the characteristics of the variables involved. A complete list of named distributions is referenced, emphasizing those limited to exponential terms. The key condition for the equality to hold is identified as independence among the variables. Overall, the conversation highlights the relationship between independence and the factorization of probability 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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