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From: Ron Baker, Pluralitas! on 19 Jun 2005 13:43

"Kenneth T. Onyee" <kentonyee(a)hotmail.com> wrote in message
news:1119201866.251158.217870(a)g43g2000cwa.googlegroups.com...
> Very interesting. What is the definition of "acor"?
> Kenneth
>

<unsnipping>
>>
>>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

It can be inferred from the above but I could have been
explicit. acor() is autocorrelation.
acor(Z) = cor(Z,Z) = E[ Z*Z ] = E[ Z^2 ]

I'll also add
zBar = E[ Z ]
so
xyBar = E[ XY ] = cor(X,Y)

>>If X and Y independent
>>
>> = acor(X) * acor(Y) - xBar^2 * yBar^2
>>

--
rb


From: Stephen J. Herschkorn on 19 Jun 2005 14:49
Ron Baker, Pluralitas! wrote:

>"Kenneth T. Onyee" <kentonyee(a)hotmail.com> wrote in message
>news:1119201866.251158.217870(a)g43g2000cwa.googlegroups.com...
>
>
>>Very interesting. What is the definition of "acor"?
>>Kenneth
>>
>>
>>
>
><unsnipping>
>
>
>>>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
>>>
>>>
> <>
> It can be inferred from the above but I could have been
> explicit. acor() is autocorrelation.
> acor(Z) = cor(Z,Z) = E[ Z*Z ] = E[ Z^2 ]
>
> I'll also add
> zBar = E[ Z ]
> so
> xyBar = E[ XY ] = cor(X,Y)


Well, first of all, correlation is defined by

corr(x, y) = Cov(X, Y) / [sqrt(Var(X)) sqrt(Var(Y))],

where Cov(X,Y) = E[(X- EX) (Y-EY)] = E[XY] - EX EY,

so your definitions are quite nonstandard. You seemed to have assumed
zero mean and unit variance.

Also, understanding your notation, your formula is basically just a
change of notation from the definition of variance:

Var(XY) = E[(XY)^2] - E[XY]^2.

which, in this case, we could simplify as

E[(XY)^2] - E[X]^2 E[Y]^2

and which doesn't help the OP much.


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

From: Ron Baker, Pluralitas! on 19 Jun 2005 15:47

"Stephen J. Herschkorn" <sjherschko(a)netscape.net> wrote in message
news:42B5BE28.9060609(a)netscape.net...
> Ron Baker, Pluralitas! wrote:
>
>>"Kenneth T. Onyee" <kentonyee(a)hotmail.com> wrote in message
>>news:1119201866.251158.217870(a)g43g2000cwa.googlegroups.com...
>>
>>>Very interesting. What is the definition of "acor"?
>>>Kenneth
>>>
>>>
>>
>><unsnipping>
>>
>>>>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
>>>>
>> <>
>> It can be inferred from the above but I could have been
>> explicit. acor() is autocorrelation.
>> acor(Z) = cor(Z,Z) = E[ Z*Z ] = E[ Z^2 ]
>>
>> I'll also add
>> zBar = E[ Z ]
>> so
>> xyBar = E[ XY ] = cor(X,Y)
>
>
> Well, first of all, correlation is defined by
>
> corr(x, y) = Cov(X, Y) / [sqrt(Var(X)) sqrt(Var(Y))],

I see that defined as the correlation *coefficient*.
http://mathworld.wolfram.com/CorrelationCoefficient.html

>
> where Cov(X,Y) = E[(X- EX) (Y-EY)] = E[XY] - EX EY,
>
> so your definitions are quite nonstandard.

See: Alberto Leon-Garcia, "Probability and Random Processes
for Electrical Engineering", second editition, page 233.
"... it is customary to call E[XY], the correlation of X
and Y." Then at the bottom of the page he defines
the _correlation coefficient_ as you describe.

> You seemed to have assumed zero mean and unit variance.

Negatory.

>
> Also, understanding your notation, your formula is basically just a
> change of notation from the definition of variance:
>
> Var(XY) = E[(XY)^2] - E[XY]^2.

That's about right. (Except I don't see that
as the 'definition' of variance but rather a useful
corollary or rearrangement.) I included the intermediate steps
but if one is already comfortable with them then,
sure, it saves time to skip them.

>
> which, in this case, we could simplify as
>
> E[(XY)^2] - E[X]^2 E[Y]^2

X and Y uncorrelated? (That isn't right unless
they are uncorrelated.) In which case it is also:
E[X^2]E[Y^2] - E[X]^2 E[Y]^2

>
> and which doesn't help the OP much.

OP? original poster?
OK. Well I guess that is for him to decide.

--
rb


From: Stephen J. Herschkorn on 19 Jun 2005 16:18
Ron Baker, Pluralitas! wrote:

>"Stephen J. Herschkorn" <sjherschko(a)netscape.net> wrote in message
>news:42B5BE28.9060609(a)netscape.net...
>
>
>>Ron Baker, Pluralitas! wrote:
>>
>>
>>
>>>"Kenneth T. Onyee" <kentonyee(a)hotmail.com> wrote in message
>>>news:1119201866.251158.217870(a)g43g2000cwa.googlegroups.com...
>>>
>>>
>>>
>>>>Very interesting. What is the definition of "acor"?
>>>>Kenneth
>>>>
>>>>
>>>>
>>>>
>>><unsnipping>
>>>
>>>
>>>
>>>>>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
>>>>>
>>>>>
>>>>>
>>><>
>>>It can be inferred from the above but I could have been
>>>explicit. acor() is autocorrelation.
>>>acor(Z) = cor(Z,Z) = E[ Z*Z ] = E[ Z^2 ]
>>>
>>>I'll also add
>>>zBar = E[ Z ]
>>>so
>>>xyBar = E[ XY ] = cor(X,Y)
>>>
>>>
>>Well, first of all, correlation is defined by
>>
>>corr(x, y) = Cov(X, Y) / [sqrt(Var(X)) sqrt(Var(Y))],
>>
>>
>
>I see that defined as the correlation *coefficient*.
>http://mathworld.wolfram.com/CorrelationCoefficient.html
>
>
>
>>where Cov(X,Y) = E[(X- EX) (Y-EY)] = E[XY] - EX EY,
>>
>>so your definitions are quite nonstandard.
>>
>>
>
>See: Alberto Leon-Garcia, "Probability and Random Processes
>for Electrical Engineering", second editition, page 233.
>"... it is customary to call E[XY], the correlation of X
>and Y."[...]
>
>

That is completely nonstandard. My specialty is applied probablity, and
I have *never* seen correlation defined that way. I challenge you to
find another reference that defines it as such. My definition is the
standard. See any standard probability text, e.g., by Ross,
Billingsley, or Feller. Even mathworld does not define correlation as
does your source.

>
>
>> You seemed to have assumed zero mean and unit variance.
>>
>>
>
>Negatory.
>
>
>
>>Also, understanding your notation, your formula is basically just a
>>change of notation from the definition of variance:
>>
>>Var(XY) = E[(XY)^2] - E[XY]^2.
>>
>>
>
>That's about right. (Except I don't see that
>as the 'definition' of variance but rather a useful
>corollary or rearrangement.) I included the intermediate steps
>but if one is already comfortable with them then,
>sure, it saves time to skip them.
>
>
>
>>which, in this case, we could simplify as
>>
>>E[(XY)^2] - E[X]^2 E[Y]^2
>>
>>
>
>X and Y uncorrelated? (That isn't right unless
>they are uncorrelated.) In which case it is also:
> E[X^2]E[Y^2] - E[X]^2 E[Y]^2
>
>

The OP said off the bat that X and Y are uncorrelated. However, that
does *not* imply that X^2 and Y^2 are uncorrelated. For an example:

P{X = 0} = 1/2.
X has conditional Uniform(-1,1) distribution given X != 0.
Y has conditional Uniform(-1,1) distribution given X = 0.
P(Y = 0 | X != 0) = 1.

--
Stephen J. Herschkorn sjherschko(a)netscape.net
Math Tutor in Central New Jersey and Manhattan
From: Kenneth T. Onyee on 19 Jun 2005 16:46
I am the OP. Let me re-state my problem. It is:
Assume Cov[X,Y] = 0. What is Cov[X Y, X Y]?

I am not interested in "correlation." I do not make any additional
assumptions
about independence -- only that Cov[X,Y]=0. Is there a simple, general
formula for the Var[X Y] under these circumstances?

Onyee

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