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This is a rather trivial question but I still hope that it is ok. In classes we got to use very often that if two random variables $X$ and $Y$ are independent, then also for example $cX,cY$ for some constant $c>0$ or $d=1/\sqrt{c}$ and others. Also $e^X, e^Y$ and so on. I could not find properties in our lecture notes or in wikipedia, but I am sure that this has to be some well established result, I just wanted to ask about the name maybe and where to find it (proofs). I am not sure if this is just for continous functions or more.

Thank you in advance! :)

AnTheBan
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1 Answers1

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If $X$ and $Y$ are independent random variables and $f,g: \mathbb R \to \mathbb R$ are Borel measurable functions then $$P(f(X) \in A, g(X)\in B)=P(X\in f^{-1} (A), Y \in g^{-1}(B))$$ $$=P(X\in f^{-1} (A))P(Y \in g^{-1}(B))=P(f(X) \in A) P(g(X) \in B).$$ Hence $f(x)$ and $g(Y)$ are independent. In particular this is true if $f$ and $g$ are continuous.

Kavi Rama Murthy
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