Re: Independent random variables versus non correlated variables
From: Aleks Jakulin (a_jakulin_at_@hotmail.com)
Date: 11/16/04
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Date: Tue, 16 Nov 2004 12:52:09 +0100
Will:
>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.
>
> Now, in Statistics we talk about variables being independent, which
> means among other things that they are not correlated. I can't see
> 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?
Each measurement/experiment/instance 'i' can be seen as having or
being characterized by two attributes/random variables/random
quantities X_i and Y_i. Now, there are many different "events" here:
if the range of X_i is {x1,x2,x3...}, and the range of Y_i is
{y1,y2,y3,...}, a possible event is any combination of x_j and y_k.
For example, if the range of your X is {sick,healthy} and the range of
your Y is {white-tongue, red-tongue}, an event will be a healthy
patient with a white tongue.
For each combination of two values x_j and y_k, you can assess the
dependence/independence through querying P(x_j,y_k) ?= P(x_j)P(y_k).
The important difference is that in probability theory P is given, but
in statistics, P must be inferred or estimated from data. Therefore,
any statement about the dependence and independence of two variables
depends on the model you choose for P. In the above case with discrete
variables, you could employ a Multinomial model. Furthermore, you have
to assess the dependence/independence for all combinations of the
variables' values.
-- mag. Aleks Jakulin http://www.ailab.si/aleks/ Artificial Intelligence Laboratory, Faculty of Computer and Information Science, University of Ljubljana, Slovenia.
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