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This question already has an answer here:

If $X$ and $Y$ are independent,

  • can we assume $X^2$ and $Y^2$ are independent in general?
  • can we assume $X^2$ and $Y^2$ are independent if $X$ and $Y$ are also identically distributed (in addition to being independent)?
  • can we assume $X^2$ and $Y^2$ are independent if both $X$ and $Y$ (which are assumed independent) followed the normal distribution (not necessarily identically distributed)?
  • can we assume $X^2$ and $Y^2$ are independent if both $X$ and $Y$ followed the normal distribution with identical mean and variance (in addition to being independent)?
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marked as duplicate by whuber Sep 22 '16 at 20:33

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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In general, $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent for any measurable functions $f,g$. Since $f(X) = X^2$ and $g(Y) = Y^2$ are measurable, the squares are independent (if $X$ and $Y$ were independent).

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