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From: Kenneth T. Onyee on 18 Jun 2005 23:18
Let X and Y be uncorrelated random variables.
By construction, this means Cov[X,Y] = 0.
Is there a simple expression for Var[XY]?
Or for Cov[XY,X]?

(I don't imagine that Var[XY] = Var[X] * Var[Y] + ???)

Kenneth

From: Stephen J. Herschkorn on 18 Jun 2005 23:43
Kenneth T. Onyee wrote:

>Let X and Y be uncorrelated random variables.
>By construction, this means Cov[X,Y] = 0.
>Is there a simple expression for Var[XY]?
>Or for Cov[XY,X]?
>
>

Off-hand, I think no. If X and Y are independent, there is a simple
expression. But its derivation uses independence in an essential way,

--
Stephen J. Herschkorn sjherschko(a)netscape.net
Math Tutor in Central New Jersey and Manhattan

From: Ron Baker, Pluralitas! on 19 Jun 2005 03:33

"Kenneth T. Onyee" <kentonyee(a)hotmail.com> wrote in message
news:1119151127.412846.142900(a)g44g2000cwa.googlegroups.com...
> Let X and Y be uncorrelated random variables.
> By construction, this means Cov[X,Y] = 0.
> Is there a simple expression for Var[XY]?


var[XY] = E[ (XY - xyBar)^2 ]
= E[ (XY)^2 - XYxyBar - XYxyBar + xyBar^2 ]
= E[ (XY)^2 ] - 2*xyBar*E[XY] + xyBar^2
= E[ (XY)^2 ] - xyBar^2
= E[ (XY)^2 ] - E[XY]^2
= acor(XY) - cor(X,Y)^2

note that 'cor' is correlation, not correlation coefficient.

If X and Y independent

= acor(X) * acor(Y) - xBar^2 * yBar^2

> Or for Cov[XY,X]?


Similar steps as above can be applied here but
my attention span has been exceeded. ;)

>
> (I don't imagine that Var[XY] = Var[X] * Var[Y] + ???)
>
> Kenneth
>

--
rb


From: Kenneth T. Onyee on 19 Jun 2005 13:24
Very interesting. What is the definition of "acor"?
Kenneth

From: Stephen J. Herschkorn on 19 Jun 2005 13:26
Ron Baker, Pluralitas! wrote:

>"Kenneth T. Onyee" <kentonyee(a)hotmail.com> wrote in message
>news:1119151127.412846.142900(a)g44g2000cwa.googlegroups.com...
>
>
>>Let X and Y be uncorrelated random variables.
>>By construction, this means Cov[X,Y] = 0.
>>Is there a simple expression for Var[XY]?
>>
>>
>
>
>var[XY] = E[ (XY - xyBar)^2 ]
> = E[ (XY)^2 - XYxyBar - XYxyBar + xyBar^2 ]
> = E[ (XY)^2 ] - 2*xyBar*E[XY] + xyBar^2
> = E[ (XY)^2 ] - xyBar^2
> = E[ (XY)^2 ] - E[XY]^2
> = acor(XY) - cor(X,Y)^2
>
>note that 'cor' is correlation, not correlation coefficient.
>
>If X and Y independent
>
> = acor(X) * acor(Y) - xBar^2 * yBar^2
>

What is acor?

--
Stephen J. Herschkorn sjherschko(a)netscape.net
Math Tutor in Central New Jersey and Manhattan

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