Regression coefficient

A coefficient of an independent variable in a regression equation. For example, in the linear regression equation ${\mathsf E} ( Y \mid X = x ) = \beta _ {0} + \beta _ {1} x$, connecting the random variables $Y$ and $X$, the regression coefficients $\beta _ {0}$ and $\beta _ {1}$ are given by

$$\beta _ {0} = m _ {2} - \rho \frac{\sigma _ {2} }{\sigma _ {1} } m _ {1} ,\ \ \beta _ {1} = \rho \frac{\sigma _ {2} }{\sigma _ {1} } ,$$

where $\rho$ is the correlation coefficient of $X$ and $Y$, $m _ {1} = {\mathsf E} X$, $m _ {2} = {\mathsf E} Y$, $\sigma _ {1} ^ {2} = {\mathsf D} X$, and $\sigma _ {2} ^ {2} = {\mathsf D} Y$. The calculation of estimates for regression coefficients (sample regression coefficients) is a fundamental problem of regression analysis.