2 Deleted cross-post on Math.SE

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

# 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.