Correlation of X with XY ?

From: Charles Knapp (nowhere_at_nomailspam.com)
Date: 09/24/04 

Date: Fri, 24 Sep 2004 14:03:47 -0400

The Pearson correlation of X with (X+Y) is
well known to be .707 [sqrt(2)]

because Var(X+Y) = Var(X) +Var(Y)
(assuming if X & Y are uncorrelated)
Hence sigma(X+Y)=sqrt(2)

         1+0
so r=----------- = 1/sqrt(2) =.707
      sqrt(2) * 1

X and Y of course are as usual, "uncorrelated
normally distributed random variables" with
zero mean and unit std. deviation.

What I want to know is what is the
correlation of X with XY ?

How do you calculate such a thing?

Or say the correlation of X with XYZ
where all 3 are "normally distributed
random variables with zero mean and unit
std. deviation"?

Now, there IS and answer to this, since
certainly it can be numerically calculated
on a computer using a random number
generator.
But how would you derive the answer mathematically?

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