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I know this is probably a very simple question, but I recall learning that for 2 random vectors, x & y, with mean mx & my,

E[(x-mx)(y-my)'] is the covariance & E[xy'] is the correlation

However, according to wikipedia, the correlation is actually: E[(x-mx)*(y-my)']/(std(x)*std(y))

Thus, what is the correct term for: E[x*y'] (if such a term exists)

Thanks!

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    $\begingroup$ The answers below are right, but I've also seen that alternative definition of "correlation" as $E[x y']$ in some areas (signal processing?). $\endgroup$ – leonbloy Nov 28 '14 at 11:58
  • $\begingroup$ @leon The alternative is correct when both $x$ and $y$ have unit ($L^2$) norms. It is equivalent to the Wikipedia formula then. $\endgroup$ – whuber Nov 28 '14 at 14:17
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$\mathbb{E}(XY)$ is known as a joint moment or mixed moment of $X$ and $Y$.

The equivalent central moment $\mathbb{E}\big((X-\mu_X)(Y-\mu_Y)\big)$ is the covariance. This will be the same as $\mathbb{E}(XY)$ if the variables have been centered so that $\mu_X = \mu_Y = 0$.

The correlation of two variables is their covariance divided by the product of their standard deviations. If the variables have unit variance, so $\sigma_X = \sigma_Y = 1$, then their covariance is the same as their correlation.

If the variables have been standardised so they have mean zero and standard deviation one, then $\mathbb{E}(XY)$ is a simple formula for the correlation. This is one of the advantages of standardising your variables, if possible, before trying to prove something about their correlation.

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The definition from wikipedia is correct. For $n$ dimensional random column vectors $x$ and $y$, the covariance matrix is n times n and it's $E[ (x-\mu_x)(y-\mu_y)^T ]$. If the means are zero, this equals $E[ xy^T ]$ and if the standard deviations are 1, this is also the correlation.

In my experience, $E[ xy^T ]$ is usually of interest only because it is the covariance, ie because at least one mean is zero.

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  • $\begingroup$ so there is no specific name for it? $\endgroup$ – DankMasterDan Nov 27 '14 at 20:24
  • $\begingroup$ When it is not the covariance, no, not as far as I know. $\endgroup$ – CloseToC Nov 27 '14 at 20:29
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So, after further research, I realized that there are indeed 2 definitions, depending on the field.

In statistics, the correlation is defined as: $\frac{E[ (x-\mu_x)(y-\mu_y)^T]}{\sigma_x\sigma_y}$

In signal processing & random processes, correlation ussually means: $E[x'y]$

As pointed out, the above quantity in statistics is refered to as the 'joint moment' or 'mixed moment'

Source: wikipedia: http://en.wikipedia.org/wiki/Autocorrelation

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