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Suppose that $X$ and $Y$ are independent discrete random variables. Prove that for any functions $f,g:\mathbb{R} \rightarrow \mathbb{R}$, the random variables $f(X)$ and $g(Y)$ are also independent.

I know I want to show that $$\mathbf{P}(f(X) = x, g(Y)=y) = \mathbf{P}(f(X) = x) \cdot \mathbf{P}(g(Y)=y),$$ but can't seem to figure out how to go about this given the fact that $X$ and $Y$ are independent.

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1 Answer 1

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$\Pr(f(X)=x)$ is the same as $\Pr(X\in f^{-1}(x))$, where $f^{-1}(x)$ does not presuppose that the function $f$ is one-to-one and so has an inverse function, but instead $f^{-1}(x)$ means the set $\{w : f(w) = x\}$.

So $\Pr(X\in\text{some set and }Y\in\text{some other set})$ can be written as a product if $X$ and $Y$ are independent. That will get you the proposed equality.

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