Is it true that it is not possible for functions of dependent random variables to be independent?
For example, if $X_1, ..., X_n$ are dependent, then it is impossible for $Y_1 = X_1^2, ..., Y_n = X_n^2$ to be independent ?
2026-03-28 12:32:14.1774701134
Example of independence of functions of dependent random variables
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Actually it is possible. Let $X$ be a random variable with the distribution $P(X=1)=P(X=-1)=\frac{1}{2}$, and let $Y=-X$. Then $X,Y$ are obviously dependent. However, $X^2$ and $Y^2$ are both constant random variables, so they are independent.