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I don't understand the last line in the following paragraph.

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How does one get that from the equation 4.3.7 using the 3 axioms of probability, which are mentioned in the book as follows:

enter image description here

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    $\begingroup$ To maximize the probability of getting an answer, please consider making the following edits to your question: 1) make it self contained: i.e. write our the equivalence you want to show (keep the reference though). 2) Change the title to describe the question rather than less relevant circumstances. 3) use latex: put your equations inside $ symbols. there \in and \leq will be useful (if you can't do that I will edit your question to fix it). 4) add other relevant tags. $\endgroup$
    – Winkelried
    Apr 7, 2018 at 13:12
  • $\begingroup$ @Winkelried I'm adding the image within the main text itself, that should make it self contained I guess? $\endgroup$ Apr 7, 2018 at 13:19
  • $\begingroup$ Since this is about a textbook and basic probability theory, I would suggest you adding the self-study tag. In that spirit, please indicate what bothers you with the derivation, like is it "if" or "only if". $\endgroup$
    – Xi'an
    Apr 7, 2018 at 16:26

2 Answers 2

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Technically, that definition of independence is incorrect. Specifically the part:

for any [my emphasis] two sets of real numbers $A$ and $B$...

Because in general one cannot consider $P[X \in A]$ for any real subset $A$. The book even adds a footnote in the first page of the chapter on continuous random variables about this.

For technical reasons one only works with sets $A\subset \mathbb{R}$ that are Borel sets. This means that the quoted part should read:

for any two real Borel sets $A$ and $B$...

This is important to your problem because it turns out that every Borel set can be build starting out with only sets of the form $\{r \in \mathbb{R}:r\leq c\}$, $c\in\mathbb{R}$ and applying finitely many times the operations of complement and countable unions. Actually, that characterizes the Borel sets of $\mathbb{R}$.

So, to show the equivalence of both conditions first note that the second condition is a special case of condition (4.3.7). The key idea for the other direction is: Since the sets $A$ and $B$ are Borel sets they can be written in therms of sets of the form $\{r \in \mathbb{R}:r\leq c\}$, $c\in\mathbb{R}$ as described above. Using the axioms, the probability operator can be distributed over that expression, you then can apply the weaker condition, and finally combine the probabilities again. To do this, consider the intermediate condition: For any Borel set $A\subset\mathbb{R}$, $b\in \mathbb{R} $: $$P\{X\in A, Y \leq b\} = P\{X\in A\}P\{Y \leq b\}.$$ Then go from your second condition to this intermediate condition and from there to (4.3.7).

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First, Ross's condition (4.3.7) is overly broad -- $A$ and $B$ need to be restricted to be any two measurable sets of real numbers, not arbitrary sets of real numbers -- but is acceptable at the level of discourse of the book where the reader is not expected to know the difference.

Next, the last sentence in the snippet glosses over the length of the proof that the condition in the definition is equivalent to $P(X\leq a, Y\leq b) = P(X\leq a)P(Y\leq b)$ for all real numbers $a$ and $b$. What is easy to show is that if $X$ and $Y$ are independent as per (4.3.7), then it follows that $P(X\leq a, Y\leq b) = P(X\leq a)P(Y\leq b)$: we simply set $A= (-\infty,a]$ and $B=(-\infty, b]$ and apply (4.3.7). (The fastidious should note that $A$ and $B$ are measurable sets of real numbers and no corners are being cut here). The reverse implication gets us into measure theory and the like.

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