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I am trying to come up with a conditional probability distribution $P(B|A)$, over the continuous variables which look something like this:

A    B    
----------
2    5    
5    7    
8    10   
10   15
12   25 
..........

etc.

How do I go about finding what distribution will fit my case?

From what I understand, it is a 2-step process. First is that I try to find what would be a good distribution to use to model this and then estimate the parameters assuming that distribution which can then be used in my calculations.

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The first step would be to determine what you want to use that distribution for. That way you can decide what features of that distribution are most important. What you ask for is a model of the data, so a trade-off between different errors is inevitable because all models are wrong.

After that I would stare at graphs, graphs, and more graphs (scatter plots, CDFs histograms and kernel density plots of A at different bins of B, etc. etc.)

Than I would choose a (set of) model(s), fit it

Than I would stare at graphs again, this time comparing the fit of the model with the data.

For both episodes of staring you need that very first step to focus your attention on characteristics that are relevant for your problem.

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