Regression Model with (Y,X) non-random? In regression, we assume that $(X,Y)$ are random variables following some certain distribution.  How would the problem change if we do not assume $(X,Y)$ are randoms. Why can we just have $Y=f(X,\epsilon)$, where $(X,Y)$ are non-random, and $\epsilon$ is a random quantity?? 

Regression has nothing to do with randomness. Regression means fitting some parametrized function or curve to some points.
That means to set values for the parameters and come up with some metric that describes the function or curve to fit better or worse than other functions or curves derived from other parameters. The result is the set of parameters that describe a function or curve that fits best according to the used metric.
It's not the concern of regression if these points are "random" or not.
