With \alpha,\beta, \sigma known in the linear relationship, can the correlation in the scatter plot of *Y* against *X* be estimated from the linear formula?

You may recall in Hierarchical Linear Model class, the scopes of the *W* dramatically impact the regression coefficients of *F*~*W* in the following R demo (hlm.jpg). While this time the regression coefficient has been fixed to a known \beta. So the scopes of *X* would never impact the regression coefficient. However, it proved that the correlation *r* could range from zero to unit (or -1) according to the variance of *X* in the final close form r=\frac{\beta\mbox{Var}\left(X\right)}{\mbox{Std}\left(Y\right)\mbox{Std}\left(X\right)}=\beta\frac{\mbox{Std}\left(X\right)}{\sqrt{\beta^{2}\mbox{Var}\left(X\right)+\sigma^{2}}}.

Let me quote as the final words from Cohen (1994; p.1001; Where the role of IV is replaced by that of DV within typical contexts like ANOVA) --

... standardized effect size measures, such as

dandf, developed in power analysis (Cohen, 1988) are, like correlations, also dependent on population variability of the dependent variable and are properly used only when that fact is kept in mind.

--

Cohen, J. (1994). The earth is round (*p*<.05). *American Psychologist*,* 49*, 997-1003.

--

Compare to the following case: different corr(s) of different IV scopes with hierarchical regression coefficients --