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If X and Y are two independent random variables and g is some function, is there some theorem saying that g(X) and g(Y) are also independent. Maybe under certain conditions for g? Monotonic? I have the feeling this should be true... under certain conditions maybe.

Also any reference or idea for a proof?

peter.petrov
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  • https://math.stackexchange.com/q/443659/321264, https://math.stackexchange.com/q/8742/321264 – StubbornAtom May 07 '20 at 20:52
  • @StubbornAtom what about for more than 2 variables in elementary probability? https://math.stackexchange.com/questions/3944284/prove-that-for-independent-random-variables-x-i-we-have-f-ix-i-are-indepe – BCLC Dec 18 '20 at 00:20

1 Answers1

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In general if $X$ and $Y$ are independent random variables and $f,g:\mathbb R\to\mathbb R$ are Borel-measurable functions then $f(X)$ and $g(Y)$ are also independent random variables.

Observe that:$$P\left(f(X)\in A,g(Y)\in B\right)=P\left(X\in f^{-1}(A),Y\in g^{-1}(B)\right)=$$$$P\left(X\in f^{-1}(A)\right)P\left(Y\in g^{-1}(B)\right)=P\left(f(X)\in A\right)P\left(g(Y)\in B\right)$$where the second equality is based on independence of $X$ and $Y$.

This proves directly what is stated: $f(X)$ and $g(Y)$ are also independent.

No further conditions on $f$ and $g$ are needed, and of course this also works if $f=g$.

drhab
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  • Thanks. What do you denote by $ g^{-1}$ here? Doesn't that imply g should be a bijection (in order to have an inverse function) – peter.petrov Dec 21 '18 at 23:06
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    In this context $f^{-1}(A)$ denotes the preimage of set $A$ under function $f$. That is: $f^{-1}(A):={x\in\mathbb R\mid f(x)\in A}$. It is not the notation of inverse function. The statements $x\in f^{-1}(A)$ and $f(x)\in A$ are equivalent statements. – drhab Dec 21 '18 at 23:08
  • Aha... OK... I kind of suspected this. Perfect. Thanks again. So it's a trivial theorem really. – peter.petrov Dec 21 '18 at 23:09
  • You are welcome. It is indeed trivial. – drhab Dec 21 '18 at 23:10
  • drhab what about for more than 2 variables in elementary probability? https://math.stackexchange.com/questions/3944284/prove-that-for-independent-random-variables-x-i-we-have-f-ix-i-are-indepe – BCLC Dec 18 '20 at 00:20
  • Is it true if we define random variable X and Y to be independient if $X^{-1}((-\infty,x]) $ and $Y^{-1}((-\infty,y]) $ are independient for all x and y? – Sikora Oct 21 '24 at 10:58
  • @Sikora Yes that is sufficient. If for all $x,y$ these sets are independent then $X$ and $Y$ are indendent random variables. – drhab Oct 21 '24 at 13:57