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I want to break down this statement:

$|E[(X - \bar{x})(Y - \bar{y})]|^2 = |<X - \bar{x}, Y - \bar{y}>|^2$

I am not familiar with expectation values being broken down into vectors. I only know that by definition $\displaystyle E[(X - \bar{x})^2] = \sum_{i=1}^{n} \frac{(x_{i} - \bar{x})^2}{n}$ and I would like to know how expectation values can be viewed as vectors specifically in the context of inner products like $E[(X - \bar{x})^2] = <X-\bar{x}, X-\bar{x}>$ Also whatever happened to the n?

My other question is how do I view covariance as a vector? I know that covariance is $E[XY] - E[X]E[Y]$ so how do I rewrite that in vector form?

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The "by definition" equality you write does not hold.

$$\displaystyle E[(X - \bar{x})^2] = \int_{S_X}(x - \bar{x})^2f_X(x) dx $$ is the correct definition for continuous r.v.'s , with $S_X$ the support of $X$ and $f_X(x)$ the pdf of $X$. For discrete random variables

$$E[(X - \bar{x})^2] = \sum_{S_X}(x - \bar{x})^2p_X(x) $$ Now IF the $x$'s can be viewed as realization of the same ergodic and stationary stochastic process, THEN $\frac {1}{n}\sum_{i=1}^{n} (x_{i} - \bar{x})^2$ is a consistent estimator of $E[(X - \bar{x})^2]$.

The expected value operator is applied to each element of any vector-matrix. If

$$A=\left[\begin{matrix} a_{11} &...& a_{1n}\\ ... & ...& ... \\ a_{k1} &...&a_{kn} \end{matrix}\right]$$

then $$E(A) = \left[\begin{matrix} E(a_{11}) &...& E(a_{1n})\\ ... & ...& ... \\ E(a_{k1}) &...&E(a_{kn}) \end{matrix}\right]$$

If $\mathbf x$ and $\mathbf y$ are two $n\times 1$ column vectors, then (prime denoting the transpose) $$ \operatorname{Cov}(\mathbf x,\mathbf y) = E(\mathbf x \mathbf y') - E(\mathbf x)\Big[E(\mathbf y)\Big]'$$

This is the expression for the covariance of two random vectors. If you want the covariance matrix of two samples, look up this answer in math.SE

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  • $\begingroup$ Your answer is informative and introduced a lot of new ideas, but I'm still wondering whatever happened to the pdf or "n" in the case of discrete r.v.s when dealing with expected values expressed as vectors? I have seen $E[XY]$ expressed as $ <X,Y>$ $\endgroup$
    – Person
    Oct 21, 2013 at 2:54
  • $\begingroup$ The size of the sample $n$ is used when we are estimating the expected value from a sample. When we define the expected value the "weight" factor is the pmf itself. Indeed, the expected value is a "weighted" average, while the sample mean is an "unweighted" average (nevertheless, the latter is a consistent estimator of the former). As for the second issue, you are confused because in other scientific fields, the symbol $< >$ is used instead of the symbol $E$ -it means exactly the same thing. $\endgroup$ Oct 21, 2013 at 8:57

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