When working with QuickCheck (the framework for tests with randomised input) I encountered some interesting connections between random variables in stochastics and their implementation in Haskell. I see the following correspondence between notions in programming and in mathematics: random generator ~ random variable / probabilistic experiment result of a random generator ~ outcome of a probabilistic experiment Thus the signature rx :: (Random a, RandomGen g) => State g a can be considered as "rx is a random variable". In the do-notation the line x <- rx means that "x is an outcome of rx". In a language without higher order functions and using a random generator "function" it is not possible to work with random variables it is only possible to compute with outcomes, e.g. rand()+rand(). In a language where random generators are implemented as objects computing with random variables is possible but still cumbersome. In Haskell we have both options either computing with outcomes do x <- rx y <- ry return (x+y) or computing with random variables liftM2 (+) rx ry This means that liftM like functions convert ordinary arithmetic into random variable arithmetic. But there is also some arithmetic on random variables which can not be performed on outcomes. For example, given a function that repeats an action until the result fulfills a certain property (I wonder if there is already something of this kind in the standard libraries) untilM :: Monad m => (a -> Bool) -> m a -> m a untilM p m = do x <- m if p x then return x else untilM p m we can suppress certain outcomes of an experiment. E.g. if State (randomR (-10,10)) is a uniformly distributed random variable between -10 and 10, then untilM (0/=) (State (randomR (-10,10))) is a random variable with a uniform distribution of {-10, ..., -1, 1, ..., 10}.