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Consider two random variables. Consider that for each random variable, n observations are drawn from the population. That is, suppose x is a vector, and it contains randomly drawn observations $x_{1}$, $x_{2}$, ..., $x_{n}$. y is defined similarly. Here the bold type font (x) is used to denote that the random variable is a vector. What is the correct notation to use to denote the covariance and the expected value? In textbooks or in lectures notes, for the covariance, sometimes I see $Cov(x_{i},y_{i})$, and sometimes $Cov$(x,y). Likewise, for the expected value, sometimes I see $E(x_{i})$, and sometimes $E$(x). Which one should one prefer?

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2 Answers 2

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I don't think there is a correct notation here, it depends on the audience.

If the intended audience includes beginners, or those without much mathematical exposure, it is useful to bold (or put an arrow over) vector, to call attention to the fact that they are not simple numbers.

If the intended audience is more experienced, they are used to having may different types of mathematical objects at play, too many to call out the type of each through notation. In this case the writer assumes the reader is responsible and experienced enough to infer the type of object from context and prior statements, so much of the type specific notation is dropped. The bolding or arrowization of vectors is often the first to go, as vector are ubiquitous in all but the most elementary topics.

In your specific case, if $x_i$ are already stated to refer to the components of a vector $\mathbf{x}$, and not individual vectors themselves, then $Cov(\mathbf{x}, \mathbf{y})$ and $Cov(x_i, y_i)$ are simply different things. The first is a martix, the second is a number.

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I don't know if econometrics does something weird, but we usually reserve capital letters for random variables, and their lower-case equivalents for their realizations. e.g., $X$ is a random variable, and $x$ is a realization of that random process.

The functions $E(\cdot)$ and $Cov(\cdot, \cdot)$ operate on random variables, so seeing $E(X)$ is usually more kosher than seeing $E(x)$, but I've seen it done, so it's not that rare. (Some authors do let lowercase letters be random variables. No notations are really standard in statistics).

I also second what Matthew Drury says. If $\mathbf{X} = (X_1, ... ,X_n)$, then:

  • $E(\mathbf{X})$ is a vector
  • $E(X_i)$ is a scalar
  • $Cov(\mathbf{X, Y})$ is a matrix
  • $Cov(X_i, Y_j)$ is a scalar
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  • $\begingroup$ Suppose x denotes a vector that contains the values 1, 2, 3. The mean is 2. How do we express the expected value? E(x) or E($x_i$)? en.wikipedia.org/wiki/Expected_value suggests E(x) (it seems wikipedia considers X to mean x). Here $x_1$ = 1, $x_2$ = 2, and $x_3$ = 3. If E($x_i$) is correct, then what does E($x_1$) mean? What does $x_i$ represent? A random scalar or a series of scalars? It must be a series of scalars. Then is it not better to write E{$x_i$}? I think my discomfort is about the meaning of E($x_i$). $\endgroup$
    – Snoopy
    Jun 4, 2017 at 19:39
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    $\begingroup$ Niether. The $E$ symbol refers to the expectation of a random variable or random vector. Your example is a sample mean, for which common notations are $\bar x$ (statistician) or $\langle x \rangle$ (physics). $\endgroup$ Jun 4, 2017 at 22:29
  • $\begingroup$ Should we then admit that the notation used here en.wikipedia.org/wiki/Expected_value is not correct? It uses E[X] while considering realisations for X? $\endgroup$
    – Snoopy
    Jun 5, 2017 at 11:43
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    $\begingroup$ No - this Wiki article is correct but uses the notation $x_1, ..., x_k$ to refer to the the support of a random variable $X$, not its realizations. I agree with Matthew, that the sample mean is almost always represented with a bar. $\endgroup$ Jun 6, 2017 at 14:47
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    $\begingroup$ No, it refers to possible realizations, which is the support of a random variable. $\endgroup$ Jun 7, 2017 at 18:18

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