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Covariance is the expectation of the product of two stochastic variables. In case of two mean-free variables $X$ and $Y$, $$ \sigma_{XY} = E[X\cdot Y]. $$ This has units of the product of the variables' units.

Correlation is covariance, normalised by the product of the individual standard deviations. $$ \rho_{XY} = \frac{E[X\cdot Y]}{\sigma_X\cdot\sigma_Y}. $$ It is unitless.

Is there a name for the quantity $$ \eta_{X\to Y} = \frac{E[X\cdot Y]}{\sigma_X^2}? $$ This has units of the ratio between the two quantities. It would seem quite useful to me: this essentially gives, for a measured $X$, the expected value of the associated $Y|X$, as $$ E(Y|X) = X \cdot \eta_{X\to Y} $$ (in a simple linear regression model).

Yet $\eta$ by itself doesn't seem to have a satisfying name. In the Wikipedia article on linear regression it's described, quite awkwardly, thus:

The slope of the fitted line is equal to the correlation between $y$ and $x$ corrected by the ratio of standard deviations of these variables.

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  • $\begingroup$ Cross-posting is not much welcome. You ought to delete cross-posts of the question. $\endgroup$ – ttnphns Sep 10 '16 at 12:39
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The satisfying name is the (unstandardized) regression coefficient. It is the main quantity what regression is about and is also called a regressional parameter.

1) From linear regression textbooks we know that for simple (two-variable) linear regression of $Y$ by $X$ correlation $r_{XY}$ equals the standardized regression coefficient $\beta_X$ (= $\beta_Y$). (Standardized regression coefficient is the regression coefficient observed when both variables are standardized to zero mean and unit variance.)

2) And from the books we know that standardized and unstandardized regression coefficient are related by $b_X=\beta_X \frac{\sigma_Y}{\sigma_X}$. (And likewise, $b_Y=\beta_Y \frac{\sigma_X}{\sigma_Y}$; and $\beta_X = \beta_Y$ for simple linear regression.)

Thence, your $\frac{cov_{XY}}{\sigma_X^2} = r_{XY} \frac{\sigma_Y}{\sigma_X} = \beta_{X} \frac{\sigma_Y}{\sigma_X} = b_X$.

For simple linear regression $b_X$ is the slope of the linear dependence of $Y$ on $X$, as Wikipedia shows it.

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  • $\begingroup$ That's better, though still a bit wordy. Do you think it would be problematic if I called it (with the definition given) simply dependence? $\endgroup$ – leftaroundabout Sep 10 '16 at 12:39
  • $\begingroup$ Dependence is very vast, generic term in statistics. You could use it if you like, but you should mean specifically "linear regression coefficient (within simple regression)". You see, outside linear regression and outside simple (two-variable) regression the formulas in your question / my answer may be invalid or should be modified accordingly. $\endgroup$ – ttnphns Sep 10 '16 at 12:43
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    $\begingroup$ @leftaroundabout How about the slope? $\endgroup$ – Jake Westfall Sep 10 '16 at 12:50

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