stochastically independence

Asked May 14 '19 at 18:59

Active May 14 '19 at 18:59

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I never took stochastic courses and need a proof for this task to continue my work at another problem. Can somebody help me out?

Let $X$ and $Y$ be stochastically independent real discrete random variables and $g,h : \mathbb{R} \rightarrow \mathbb{R}$

Show that $g(X)$ and $h(Y)$ are stochstically independent.

asked May 14 '19 at 18:59

Arji

3

If $P(X \in A, Y \in B) = P(X \in A)P(Y \in B)$, then $P(g(X) \in A, h(Y) \in B) = P(X \in g^{-1}(A), Y \in h^{-1}(B)) = P(X \in g^{-1}(A)) P(Y \in h^{-1}(B))$ under appropriate assumptions. – snar May 14 '19 at 19:09
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See e.g. Are functions of independent variables also independent? – Minus One-Twelfth May 14 '19 at 19:13

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