Re: Correlation of X with XY ?
From: Charles Metz (c-metz_at_uchicago.edu)
Date: 09/24/04
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Date: Fri, 24 Sep 2004 18:53:44 -0500
Charles,
Correlation of any two random variables U and V is defined as normalized
covariance:
r(U,V) = Covar{U,V}/sqrt[Var{U}*Var{V}]
= [E{UV}-E{U}*E{V}]/sqrt[Var{U}*Var{V}].
Taking U = X and V = XY in this expression will yield the desired answer
if you notice that uncorrelated *normal* random variables are also
independent, whence:
-- E{XY} = E{X}*E{Y}
-- E{X*XY} = E{[X^2]*Y}
= E{X^2}*E{Y}
and:
-- Var{XY} = E{(XY)^2}-[E{XY}]^2
= E{X^2}*E{Y^2}-[E{X}*E{Y}]^2
The result may seem surprising, so I encourage you to think about why it
is reasonable.
Charles Metz
---------------------
Charles Knapp wrote:
> 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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