- #1

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## Main Question or Discussion Point

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

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).

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
```

y = f(X,W).