# Separation of variables for Named Probability Density Distributions

• I
• redtree
In summary, the equality of the probability density distribution P(x) with the product of individual distributions P1(x1) to Pn(xn) is determined by the independence of the variables x1 to xn. This condition is necessary and sufficient for the equality to hold. A complete list of named distributions can be found in the provided link.
redtree
TL;DR Summary
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{split}
P(\vec{x}) &= P_1(x_1) P_2(x_2) ... P_n(x_n)
\end{split}

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

Office_Shredder and pbuk

## 1. What is separation of variables for Named Probability Density Distributions?

Separation of variables is a mathematical technique used to simplify the solution of partial differential equations by separating the variables into individual equations. This method is commonly used in probability theory to solve for the probability density function of a named distribution.

## 2. What are the benefits of using separation of variables for Named Probability Density Distributions?

Separation of variables allows for the solution of complex probability density functions to be broken down into simpler, one-dimensional equations. This makes it easier to solve for the probability density function and understand the behavior of the named distribution.

## 3. How is separation of variables applied to Named Probability Density Distributions?

The first step in applying separation of variables is to identify the variables in the probability density function. These variables are then separated into individual equations, and the resulting equations are solved independently. The solutions are then combined to form the final probability density function.

## 4. What are some common Named Probability Density Distributions that use separation of variables?

Some common named distributions that use separation of variables include the normal distribution, exponential distribution, and Poisson distribution. These distributions are commonly used in statistics, finance, and other fields to model real-world phenomena.

## 5. Are there any limitations to using separation of variables for Named Probability Density Distributions?

While separation of variables is a powerful technique, it may not always be applicable to every named distribution. In some cases, the variables may not be able to be separated, or the resulting equations may be too complex to solve. In these cases, other methods may need to be used to find the probability density function.

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