# What is the term for E[x*y']

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!

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

## 3 Answers

$\mathbb{E}(XY)$ 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.

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.

• so there is no specific name for it? – DankMasterDan Nov 27 '14 at 20:24
• When it is not the covariance, no, not as far as I know. – CloseToC Nov 27 '14 at 20:29

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