Independent random variables

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". – user21436 Jan 22 '12 at 21:48
So, you're right in what you have to do! – user21436 Jan 22 '12 at 21:48
Is that really copied verbatim from the textbook? (I hope not.) – cardinal Jan 22 '12 at 21:49
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! – user21436 Jan 22 '12 at 21:53
The textbook probably does not ask this for every pair of subsets of $\mathbb R$. – Did Jan 23 '12 at 0:26