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Quoting my text book:

Two random variables $X_{1},X_{2}$ are called independent, if:

$P(X_{1}\in A_{1}, X_{2}\in A_{2}) = P(X_{1}\in A_{1})\cdot P(X_{2}\in A_{2})$

for all $A_{1},A_{2}$ where $A_{i}$ is a subset of $\mathbb{R}$.

The 'if' in the above text confuses me.

If you have two independent variables and want to find $P(X_{1}\in A_{1}, X_{2}\in A)$ can you then just find the product of $P(X_{1}\in A_{1})$ and $P(X_{2}\in A_{2})$?

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    You need to interpret the "if" in any definition as "if and only if".2012-01-22
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    So, you're right in what you have to do!2012-01-22
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    Is that really copied verbatim from the textbook? (I hope not.)2012-01-22
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    You need to have written $P(X_1 \in A_1)$ and $P(X_2 \in A_2)$. That's what cardinal points out. Recall that probability measure is defined for events and not for random variables!2012-01-22
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    Yes! It was a mistake (my fault). But thank you!2012-01-22
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    The textbook probably does not ask this for **every** pair of subsets of $\mathbb R$.2012-01-23
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    If you know they're independent and if you know their probabilities, then certainly you can do that.2012-01-23
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    @Didier Piau: Unfortunately it's quite possible that the book does say that. For example, Evans and Rosenthal, "Probability and Statistics: The Science of Uncertainty" does that.2012-01-23
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    @Robert: Thanks for this information. But let us wait the reaction of the OP, *there is still hope*...2012-01-23

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