# product of square of two independent variables [duplicate]

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)?

## marked as duplicate by whuber♦Sep 22 '16 at 20:33

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).