How to deal with different distributions when fitting a model? I would like to know how to proceed when fitting a model (lm/glm) if
a) The distributions between the dependent and the independent variables are different (y =
Normal, x = Weibull)
b) The distributions among the predictors are different (for example: y = Normal, x1 = Normal, x2 = Weibull, x3 = Weibull).
In case of a): Do we just care of the dependent variable (y) distribution? Therefore, fitting a linear model with normal distribution without doing any transformation of the independent variable (x)?
In case of b): Do we want our predictors to be in the same distribution or do we just fit a model with all the different distributions in it?
 A: The distribution of the explanatory/independent/right-hand-side/x-variables does not matter.
It is not bad to look at these distributions: I often do look at the distributions, but mainly to check for data preparation errors. For example, and age of -99 suggests to me that I forgot to correctly code for missing values. Moreover, very skewed distributions often correspond to non-linear effects. So if I see that, I take an extra sharp look at the linearity of the effect for that variable.
Moreover, the explained/dependent/left-hand-side/y-variable is in linear regression not assumed to follow the unimodal symmetric bell-shaped normal distribution we know and love. Instead, it is a normal distribution where each observations has its own mean, depending on the explanatory variables.  The resulting distribution could be bimodel, trimodel, skewed, or pretty much anything else. So there is no easy way to look at just the distribution of y and see if this is true. Instead we look at the residuals, which should follow this nice symmetric unimodel bell shape. Moreover, this assumption usually does not matter, as linear regression is remarkably robust against deviations from this assumption.
