• #### Q independence and correlation

From Cosine@21:1/5 to All on Sat Jan 18 11:36:57 2020
Hi:

If two events are statistically independent, then they are correlated.

That is, the correlation coefficient or covariance is not zero.

However, even if two events are statistically correlated, independence is not guaranteed.

A question is are there some special cases that we are sure that correlation implies independence?

Another question is what are the examples that two events are correlated but not independent?

Since our events are obtained from sampling, how do we design/choose a sampling process that produces two independent events?

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• From Rich Ulrich@21:1/5 to All on Sat Jan 18 19:55:11 2020
On Sat, 18 Jan 2020 11:36:57 -0800 (PST), Cosine <asecant@gmail.com>
wrote:

Hi:

My. You have certainly gotten all your terms backwards.
I will notate a couple where I can fix the statement.

If two events are statistically independent, then they are correlated.

No. Independent is what imples uncorrelated.

That is, the correlation coefficient or covariance is not zero.

... IS zero

However, even if two events are statistically correlated, independence is not guaranteed.

Even when /uncorrelated/, "independence" is not guaranteed.

EXAMPLE
When there is no /linear/ correlation, there still can be
a non-linear dependence. Consider the exact definition for
Y= X**2 for X symmetric around zero.

A question is are there some special cases that we are sure that correlation implies independence?
???
Another question is what are the examples that two events are correlated but not independent?
???
Since our events are obtained from sampling, how do we design/choose a sampling process that produces two independent events?

Events are independent (or not) "with regard to" such-and-so.
"Randomization" is what we use in trials, sometimes within
stratifications.

--
Rich Ulrich

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