Consider the model $ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon$.
What is the relationship between the correlation coefficients $r_{y,x_1}$, $r_{y,x_2}$ and the regression coefficients $\beta_1$ and $\beta_2$?
In particular, how to interpret a situation where a particular correlation coefficient is statistically significant but the corresponding regression coefficient is not statistically significant?
I recall reading a concept called "partial correlation coefficients". Is that in some way relevant in the above context?