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?

## Tuesday, September 17, 2013

### "Independence" in probability ^ statistics

at Tuesday, September 17, 2013

Labels: mathStat, original_content, z_oq

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