Re: Correlation of X with XY ?

From: Charles Knapp (nowhere_at_nomailspam.com)
Date: 09/27/04 

Date: Mon, 27 Sep 2004 00:08:46 -0400

"Glen" <glenbarnett@geocities.com> wrote in message
news:bd343329.0409261755.75ce0c9b@posting.google.com...
> "Charles Knapp" <nowhere@nomailspam.com> wrote in message
news:<mYOdnRJ3ifGJLcjcRVn-sg@comcast.com>...

> > [Hammond]
>
> Why did you say "Hammond" there?

oops.... I'm not "Charles Knapp" that's an internet pseudonym.
I opened a Cape Cod phone book and that's the first name I saw.
I'm so well known on the internet that people follow me around
and I find it difficult to have a private conversation with anyone,
especially about purely technical matters without being interrupted
by illiterate fans replying to my posts. They follow my name
on the Google posting histories. Sorry for accidentally blowing my cover.

>
> > My problem seems to be with "Z-scored" normally distributed variables.
> > In that case Mean=0, Variance=1 Std. Deviation=1
> > With these values we get for the Covar{X,XY} the following:
> >
> > Covar{X,XY} = [E{X*XY}-E{X}*E{XY}]
> > = E{X^2}*E{Y}-E{X}*E(XY}
> > = ZERO
> >
> > because both E{X} and E{Y} are 0. This result can't be correct,
> > since surely X Covaries with XY in a non-zero fashion.
> >
> > Where am I going wrong?
>
> The sentence where you say:
> "This result can't be correct, since surely X Covaries with XY in a
> non-zero fashion."
>
> is where you go wrong.
>
> Imagine X and Y were /perfectly/ correlated standard normal random
> variables, so that X=Y. Then Covar(X,X^2) = 0 is easily demonstrated
> (and indeed, if you plot X^2 against X you can see there's no linear
> correlation).

Um.... I'm assuming X is totally uncorrelated with Y.
In that case it appears to me that, in general, Corr(X,XY)
will not zero. However I now see that the magnitude of this
correlation in general depends on the Means and variances
of X and Y and is not a fixed (invarient) quantity. In fact
it may still be 0 for normal random variables with mean 0
and variance 1.

However, this is in contrast to the correlation of X with (X+Y) for
instance which appears to me to be an invarient = 1/sqrt(2)
for normal random variables.

>
> The variables are perfectly related, just not in a linear way (and
> correlation only measures the linear part of the relationship).
>
> When Y is less than perfectly correlated with X, the correlation
> between XY and X is still 0.

I doubt this in the case where Y is totally uncorrelated with X.
Especially since X and Y are montonically increasing functions.

>
> Glen

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