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I originally posted this question on Math.SE

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.

I originally posted this question on Math.SE

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.

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.

1
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Is there a name for the “one-sided–normalised covariance”?

I originally posted this question on Math.SE

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.