# Latest content was relocated to https://bintanvictor.wordpress.com. This old blog will be shutdown soon.

## Tuesday, September 17, 2013

### "Independence" in probability ^ statistics

I feel probability and statistics have different interpretations of Ind, which affects our intuition --

- independence in probability is theoretical. The determination of ind is based on idealized models and rather few fundamental axioms. You prove independence like something in geometry. Black or white.

- independence in statistics is like shades of grey, to be measured. Whenever there's human behavior or biological/evolution diversification, the independence between a person's blood type, birthday, income, #kids, education, lifespan .. are never theoretically provable. Until proven otherwise, we must assume these are all dependent. More commonly, we say these "random variables" (if measurable) are likely correlated to some extent.

* ind in probability problems are pure math. Lots of brain teasers and interview questions.
* ind in stats is often related to human behavior. Rare to see obvious and absolute independence

For Independence In Probability,
1) definition is something like Pr (1<X<5 | 2<Y<3) = Pr (1<X<5) so the Y values are irrelevant.

2) an equivalent definition of independence is the "product definition" -- something like P(1<X<5 AND 2<Y<3) = product of the 2 prob. We require this to be true for any 2 "ranges" of X and of Y. I find this definition better-looking but less intuitive.

You could view these definitions as  a proposition if you already have a vague notion of independence. This is a proposition about the entire population not a sample. If you collect some samples, you may actually see deviation from the proposition!?

Actually, my intuition of independence often feels unsure. I now feel those precise definitions above are more clear, concise, provable, and mathematically usable. In some cases they challenge our intuition of independence.

An Example in statistics --If SPX has risen for 3 days in a row, does it have to do with the EUR/JPY movement?

E(X*Y) = E(X)E(Y) if X and Y are independent. Is this also an alternative definition of independence? Not sure.

I feel most simple examples of independence are the probability kind -- "obviously independent" by common sense. It's not easy to establish using statistics that some X and Y are independent. You can't really collect data to deduce independence, since the calculated correlation will  likely be nonzero.

Simple example?