# Decomposition of a R.V. into dependent and independent parts

• mXSCNT
In summary, it is possible to find a random variable W which is independent from X, such that Y = f(X,W) for some function f, as long as the "=" in the equation is interpreted as "having the same distribution." However, it is not possible to find such a W and f if the "=" is interpreted as "almost surely equal," as this would generally depend on X.
mXSCNT
Given random variables X and Y, which are not independent, is it always possible to find a random variable W which is independent from X, such that Y = f(X,W), for some function f?

Example: let the joint distribution of X and Y be
Code:
 Y  0  1
X+-------
0|1/3 1/6
1|1/6 1/3
Then if we let W have the Bernoulli distribution with p = 1/3, and f(x,w) = x XOR w, we have
y = f(X,W).

yes. You can prove it as follows.

Define,
g_X(y) = P(Y<=y|X)
which gives the cumulative distribution function of Y conditional on X.
Define the variable
U=g_X(Y)
As long as g_X is continuous (so Y has continuous distribution) then U will be uniformly distributed on [0,1] independently of X.

Setting f(x,u)=g_X^{-1}(u) gives Y=f(X,U).

If g_X is not continuous then you can still set Y=f(X,U) for a r.v. U uniform on [0,1] independently of X, and X,Y will have the correct joint distribution. Note that this can require enlarging the probability space in order to introduce U.

Hopefully you can fill in the gaps in that brief proof.

mXSCNT said:
Given random variables X and Y, which are not independent, is it always possible to find a random variable W which is independent from X, such that Y = f(X,W), for some function f?

Depends on what you mean by "=" in the above equation. You can definitely find such a W and f such that f(X,W) has the same distribution as Y (in fact, you don't even need an X here at all), or even such that (X,Y) and (X,f(X,W)) have the same joint distribution.

But if by "=" you mean "almost surely equal," then, no, such a result would generally have to depend on X. I.e., you'd use W = Y-X, and f(X,W) would be X+W. But that is clearly NOT independent of X. Simply put, there's generally no way that you can replace the outcome of a random variable with the outcome of an independent random variable, and hope to get the same answer all of the time.

For the discrete case: Let A be the domain of X and B be the domain of Y. Let W be a random variable defined on BA, where P(W[x] = y) = P(Y=y | X=x). Let f(x,w) = w[x]. Then P(f(X,W) = y) = P(W[X] = y) = P(Y=y | X), which is the distribution of Y.

## What is the definition of "Decomposition of a R.V. into dependent and independent parts"?

The decomposition of a random variable (R.V.) into dependent and independent parts is a statistical technique used to break down the variability of a random variable into two components: one that is affected by other variables known as the dependent part, and one that is not affected by other variables known as the independent part.

## What is the purpose of decomposing a R.V. into dependent and independent parts?

The purpose of decomposition is to better understand the relationship between a random variable and other variables. By separating the dependent and independent parts, we can identify and quantify the extent to which other variables influence the variability of the random variable. This can help in making predictions and understanding the underlying factors driving the variability of the random variable.

## How is a R.V. decomposed into dependent and independent parts?

There are several methods for decomposing a random variable into dependent and independent parts, including ANOVA (Analysis of Variance), regression analysis, and factor analysis. These techniques use statistical models to identify and quantify the effects of other variables on the random variable.

## What are some examples of dependent and independent parts in a R.V.?

An example of a dependent part in a random variable could be the effect of temperature on plant growth, where the temperature is the independent variable and plant growth is the dependent variable. An example of an independent part could be a person's height, which is not affected by any other variables.

## What are the limitations of decomposing a R.V. into dependent and independent parts?

Decomposition of a random variable into dependent and independent parts is not always straightforward and can be influenced by various factors, such as the quality of the data and the choice of statistical model. Additionally, it does not necessarily imply causation between variables, as correlation does not always equal causation. Finally, it is important to remember that decomposition is only one aspect of understanding the relationship between variables and should be used in conjunction with other statistical techniques for a comprehensive analysis.

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