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In probability books, the definition of independent discrete random variables are often given as

The random variables $X$ and $Y$ are said to be independent if $\mathbb P(X \leq x, Y \leq y) = \mathbb P(X \leq x) \mathbb P(Y \leq y)$ for any two real numbers $x$ and $y$, where $\mathbb P(X \leq x, Y \leq y)$ represents the probability of occurrence of both event $\{X \leq x\}$ and event $\{Y \leq y\}$.

or

$\mathbb P(X \in A, Y \in B) = \mathbb P(X \in A) \mathbb P(Y \in B)$

And the 2 definitions are alleged to be identical. But the proof is often omitted. Although it's intuitively correct, I still want to see a proof. Could anyone show me how to prove this?

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The proof is often omitted? What texts are you looking in? In my experience, this is usually one of the first things proved about independent random variables. See S. Resnick (1999), A probability path, Birkhauser, pp. 91ff, or R. Durrett, Probability: Theory and Examples, 4th. ed., Cambridge, sec 2.1 for details. The proof is a nice example of using the $\pi\mbox{-}\lambda$ theorem. Resnick's treatment is a bit more careful and detailed (or, tedious, depending on your perspective) than Durrett's, but they both follow along the very same lines. –  cardinal Sep 15 '11 at 12:20
    
Maybe I haven't checked enough text books ... –  ablmf Sep 15 '11 at 14:02
    
Which one(s) have you checked? –  Did Sep 16 '11 at 10:10
    
Here are some additional references on my shelf: (1) D. Williams (1991), Probability with Martingales, Cambridge, Ch. 4. (2) A. N. Shiryaev (1996), Probability, Springer, 2nd. ed., Ch. II, Sec. 5, pp. 179ff. (3) P. Billingsley (1995), Probability and measure, 3rd. ed, Wiley, Sec. 20, pp. 263ff. (4) J. Jacod and P. Protter (2004), Probability essentials, Springer, Ch. 10. In every one of the six examples given, the proof you asked for is the very first result given in the section on independent random variables. –  cardinal Sep 18 '11 at 18:51
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1 Answer

up vote 2 down vote accepted

These two definitions are equivalent due to the following reason. I assume that you mean $A,B$ be Borel measurable. Then the 2nd definition says that $\sigma(X)$ is independent of $\sigma(Y)$ where $$ \sigma(X) = \{X^{-1}(B)|B\in\mathcal{B}(\mathbb{R})\} $$ and $$ \sigma(Y) = \{Y^{-1}(B)|B\in\mathcal{B}(\mathbb{R})\}. $$

Clearly, 2nd definitino implies the first one. To see that the first implies the second just recall that $\mathcal{B}(\mathbb R)$ is the smallest $\sigma$-algebra which includes the class $\{(-\infty,x]|x\in\mathbb R\}$.

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