"My Stats homework has a question of the type: "Given this joint distribution over X and Y, compute E(E(X|Y))."... This notation is extremely confusing.Given how unclear the notation is, I decided to do something about it, using the formal(ish?) language that I designed yesterday..." --- A(nother) interesting attempt at a "formal language with types" approach to probabilistic notation. It's good! But to be honest, I think ideas like this occur to a lot of Haskell-aware CS people when they first encounter probabalistic notation -- I remember John Barnett showing me something like this four or five years ago. I'd recommend looking at some of the papers of Avi Pfeffer and Claire Jones, if you want to see where this stuff ends up going. In particular, Pfeffer's 2000 thesis, or his IBAL paper, or his "Stochastic Lambda Calculus and Monads of Probability Distributions" paper, and the references therein. Things get interesting when you go to higher-order modeling languages...