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Is there literally any [ preferably free ] Computer Algebra System that would handle probability theory directly? As for now I am mainly interested in calculating marginal densities out of user-defined joint densities on diverse domains, but a complete system would be even better. And by any system I mean also that operating system is not an issue as long it works on one of the big three (Linux, Windows, Mac OS X).

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Recommend you have a look at: en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems (it looks liker there are three choices and two of them are commercial, but any of the free ones can do these things - just not canned in the program, you have to do work and programming in that environment). HTH –  Amzoti Sep 19 '12 at 16:45
    
@Amzoti If by doing work you mean to present an appropriate integral, then I can already do it in the CAS I currently use - namely wxMaxima. But the problem is that as for now I tend to make errors precisely in this step ( being still a student ) and among other things it is the very skill I want to practice –  infoholic_anonymous Sep 19 '12 at 16:57
    
Well, what you ask about is just doing some integrals, and you do not need ny particular "probability functionality" to do that ... –  kjetil b halvorsen Sep 19 '12 at 17:15
    
Have you tried R? –  Alexander Konovalov Jul 28 '13 at 19:01
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