One might expect there to be a formal relationship between a regression coefficient and a correlation coefficient (at least intuitively). Does this relationship change if the distribution of the data is non-gaussian (e.g., bionomial)?
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Let $\rho=Corr(X,Y)$ For a simple linear regression, $Y=\beta_0+\beta_1X +\epsilon$, you have $\beta_1=\rho \frac{\sigma_Y}{\sigma_X}$
Kamster
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@Kamster answer is correct in the univariate case.
If there are more variables then regression coefficient can be thought as a partial correlation coefficient when effects of other variables are controlled.
Analyst
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2Standardized coefficient is indeed closely related (but not equal) to partial correlation: http://stats.stackexchange.com/a/76819/3277. – ttnphns Jan 05 '15 at 12:33
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@ttnphns that is true. – Analyst Jan 05 '15 at 13:30