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Let $X$ be a random variable with expectation value $\mathbb{E}(X)=\mu$.

Is there a (reasonably standard) notation to denote the "centered" random variable $X - \mu$?

And, while I'm at it, if $X_i$ is a random variable, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, and if $\overline{X} = \frac{1}{n}\sum_{i\in\mathbf{n}} X_i$, is there a notation for the random variable $X_i - \overline{X}$? (This second question is "secondary". Feel free to disregard it.)

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Standard and widely used notations and terminologies exist in some contexts; see my answer below. – Michael Hardy Apr 7 '13 at 15:21
up vote 2 down vote accepted

In some contexts, $\varepsilon_i = X_i-\mu$ is called an "error" and $\hat\varepsilon_i=X_i-\bar X$ is called a "residual".

Notice that if $X_1,\ldots,X_n$ are i.i.d. then the errors $\varepsilon_i=X_i-\mu$ are independent and the residuals $\hat\varepsilon_i=X_i-\bar X$ are not (since they are constrained to add up to $0$, so they are negatively correlated).

In the simple linear regression problem in which $\mathbb E(X_i) = \alpha+\beta w_i$, the errors $\varepsilon_i = X_i - (\alpha+\beta w_i)$ are often taken to be independent, but the residuals $\hat\varepsilon_i=X_i - (\hat\alpha+\hat\beta w_i)$, where $\hat\alpha$ and $\hat\beta$ are least-squares estimates that depend on $X_i$ and $w_i$, $i=1,\ldots, n$, are constrained to satisfy the two equalities $\hat\varepsilon_1+\cdots+\hat\varepsilon_n=0$ and $\hat\varepsilon_1 w_1+\cdots+\hat\varepsilon_n w_n=0$. The correlation between $\hat\varepsilon_i$ and $\hat\varepsilon_j$ depends on $w_1,\ldots,w_n$ and on $i$ and $j$. (Specifically, if the errors all have the same variance---an assumption called homscedasticity---and are uncorrelated, and every entry in the first column of $M\in\mathbb R^{n\times 2}$ is $1$ and the second column is $w_1,\ldots,w_n$, then $\operatorname{var}(\varepsilon)(I_n-M(M^TM)^{-1}M^T)$ is the matrix whose $i,j$ entry is $\operatorname{cov}(\hat\varepsilon_i,\hat\varepsilon_j)$.)

See this article:

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I've never seen any specific notation for these. They are such simple expressions that there wouldn't be much to gain by abbreviating them further. If you feel you must, you could invent your own, or just say "let $Y = X - \mu$".

One way people often avoid writing out $X - \mu$ is by a statement like "without loss of generality, we can assume $E[X] = 0$" (provided of course that we actually can).

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The reason for a notation in this case would not be typographical economy, but rather to identify the "centered" variable as a sort of "residual", or "error", incurred in by using the constant $\mu$ as an approximation for the random variable $X$. – kjo Apr 7 '13 at 14:01
@NateEldredge : Read my answer to the question and you will lose your virginity. – Michael Hardy Apr 7 '13 at 15:30

Sometimes, we can encounter the notation $\overset{\circ}X$.

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Source? $ $ $ $ – Did Apr 7 '13 at 12:04
I saw it in a French book (Ouvrard, Probabilités). – Davide Giraudo Apr 7 '13 at 13:22
Thanks. The notation seems hardly universal, even when restricted to the French speaking mathematical world. – Did Apr 7 '13 at 13:29

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