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The book I am reading says the following:

For any two variables $X$ and $Y$, if for every two sets $A$ and $B$ of real numbers $Pr(X \in A \cap Y \in B) =Pr(X \in A )Pr( Y \in B)$, then $X$ and $Y$ are independent. Further, if we can show that $P(X \leq x \cap Y \leq y) = P(X \leq x)P(Y \leq y)$, then we can conclude that $X$ and $Y$ are independent random variables.

Does anybody have a proof for this last statement? My book has omitted the proof.

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  • $\begingroup$ Have you seen a proof that the distribution function determines the distribution? That would make very short work of this problem. $\endgroup$ – whuber Mar 29 at 12:37
  • $\begingroup$ @whuber I am afraid not. Could you share a link or something? $\endgroup$ – Avneesh Khanna Mar 30 at 13:03
  • $\begingroup$ A modern textbook on probability or probability and measure would include a rigorous account. Just about any textbook on probability and statistics will at least assert this result and perhaps demonstrate it in simple cases. $\endgroup$ – whuber Mar 30 at 15:21

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