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Probability

## If A,B,C are independent, then AnB (resp., AuB) and C are independent

Posted 02-02-2009 at 01:12 AM by Marid Audran

(Using n for intersection and u for union, since the built-in TeX is terrible with those symbols.)

Oddly enough, Rosenthal's probability book doesn't seem to even point this out, and my other probability book (Hogg/Craig, from several years back) doesn't see fit to prove it or make a big deal of it. I think it's a big deal. If I've got it right, then I think the proof helps explain why "independence" is defined as a distinct phenomenon from "pairwise independence."*...

## Probability theory textbook: Secret weapon or red herring?

Posted 02-01-2009 at 12:33 AM by Marid Audran

As interesting as the Rosenthal probability text is to me, I wonder whether I'm doing myself a disservice by spending much time on it. My hope is that I'll better understand the underlying theory (e.g., measure theory) and thus be better equipped to do challenging exam problems (not to mention being more ready to actually use probability in the workplace). But it's clear that others have tried to understand the deeper theory too and mostly found it unnecessary (see, for example, this thread on Ito's...

## Divergence of the harmonic series, in a probability context

Posted 01-27-2009 at 10:42 PM by Marid Audran
Updated 02-06-2009 at 11:54 PM by Marid Audran (Adding a remark that it's not obvious)

I thought of this while reading in Rosenthal's book about the Borel-Cantelli lemma and trying to think up an example.

I learned a long time ago that the harmonic series
$\sum_{n=1}^\infty \frac 1n$
diverges to infinity. Nonetheless, when I thought of the following example (which boils down to this fact), it felt less than completely intuitive to me.

Consider an infinite sequence of real numbers $x_1, x_2, x_3, ...$, where each number is chosen randomly from the interval $(0,1)$...