Detecting Different Distributions in data Supposing I have a dataset that I assume that have instances generated by two different distributions, is it possible to separate these instances based on the underlying generating distribution?  

Also, if I relax this hypothesis, and suppose I know one of the distributions beforehand, is this task possible? The second distribution could he chaotic, and also it is impossible to know it. 

Finally, is there any result concerning the distance between the two distributions? Is there any theoretical proof of correctness, if there are algorithms for realizing that? 

Thank you!
It's not really feasible to label points as coming from distribution A or B if there is be overlap amongst the distributions. You can't know whether the point fell near the mean of A or the tail of B. In the simplest case, consider two normally distributed datasets with overlap - how can you know which generated it?

You could estimate the probability of each point falling under A or B, but that doesn't really solve the problem. 

A different approach, but one I'm not sure will help, would be to fit a new distribution defined as the convolution of A and B. The ex-gaussian distribution is one such example. But you'd have to do the maths to figure out the PDF equation, and that still won't necessarily label your points.