Re: Independent random variables versus non correlated variables
From: Herman Rubin (hrubin_at_odds.stat.purdue.edu)
Date: 11/16/04
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Date: 16 Nov 2004 09:52:31 -0500
In article <1c58f047.0411160246.7bf0153c@posting.google.com>,
Will <william108@gmail.com> wrote:
>I have a question which I guess revolves around the meaning of
>independence of random variables. The definition I am aware of is in
>the context of discrete rvs X and Y where for all possible values of x
>and y, p(x,y)=p(x)p(y).
>So this means that in terms of independence of events, for each
>possible combination of events generated by X and Y those events are
>independent.
In fact, this is the way it is used, and should be the
"definition". The formula is just a means of testing.
But it should be used with the idea of probability, not
of computation.
>Now, in Statistics we talk about variables being independent, which
>means among other things that they are not correlated. I can't see how
>this relates to the above "for each possible combination of events
>generated by X and Y those events are independent".
>Can anyone give me some insight into this?
>Thanks so much.
The idea of independence is that there is no probability
information given from one (or some) about the other(s).
Two events are independent if knowing whether or not one
has occurred does not change the estimate of the
probability of the other, except with probability zero.
Two random variables are independent if, except with
probability zero, given any information about one does not
affect the assessment of the other. An indexed family of
random variables is independent if knowing what happened
for one set of indices does not change the opinion about
those with the rest of the indices.
Being not correlated means, if variances exist, that
E(XY) = E(X)E(Y). Now always if variances exist,
E(XY) = E(X(E(Y|X))). Form independence, E(Y|X) = E(Y),
which is a constant as far as X is concerned, and this
gives the lack of correlation.
To summarize, independence of a set of random variables
means that knowing something about some of them does
not change the probability distribution of the others.
The formulas are just means of reducing how much has to
be done to check this.
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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