A simple monad transformer to allow computations in the transformed monad to generate random values.

## The code

```{-# OPTIONS_GHC -fglasgow-exts #-}

getRandom,
getRandomR,
getRandoms,
getRandomRs,
evalRandT,
evalRand,
evalRandIO,
fromList,
Rand, RandT -- but not the data constructors
) where

import System.Random
import Control.Arrow

getRandom :: (Random a) => m a
getRandoms :: (Random a) => m [a]
getRandomR :: (Random a) => (a,a) -> m a
getRandomRs :: (Random a) => (a,a) -> m [a]

newtype (RandomGen g) => RandT g m a = RandT (StateT g m a)

liftState :: (MonadState s m) => (s -> (a,s)) -> m a
liftState t = do v <- get
let (x, v') = t v
put v'
return x

getRandom = RandT . liftState \$ random
getRandoms = RandT . liftState \$ first randoms . split
getRandomR (x,y) = RandT . liftState \$ randomR (x,y)
getRandomRs (x,y) = RandT . liftState \$
first (randomRs (x,y)) . split

evalRandT :: (Monad m, RandomGen g) => RandT g m a -> g -> m a
evalRandT (RandT x) g = evalStateT x g

runRandT  :: (Monad m, RandomGen g) => RandT g m a -> g -> m (a, g)
runRandT (RandT x) g = runStateT x g

newtype Rand g a = Rand (RandT g Identity a)

evalRand :: (RandomGen g) => Rand g a -> g -> a
evalRand (Rand x) g = runIdentity (evalRandT x g)

runRand :: (RandomGen g) => Rand g a -> g -> (a, g)
runRand (Rand x) g = runIdentity (runRandT x g)

evalRandIO :: Rand StdGen a -> IO a
evalRandIO (Rand (RandT x)) = getStdRandom (runIdentity . runStateT x)

fromList :: (MonadRandom m) => [(a,Rational)] -> m a
fromList [] = error "MonadRandom.fromList called with empty list"
fromList [(x,_)] = return x
fromList xs = do let s = fromRational \$ sum (map snd xs) -- total weight
cs = scanl1 (\(x,q) (y,s) -> (y, s+q)) xs -- cumulative weight
p <- liftM toRational \$ getRandomR (0.0,s)
return \$ fst \$ head \$ dropWhile (\(x,q) -> q < p) cs
```

To make use of common transformer stacks involving Rand and RandomT, the following definitions may prove useful:

```instance (MonadRandom m) => MonadRandom (StateT s m) where
getRandom = lift getRandom
getRandomR r = lift \$ getRandomR r

getRandom = lift getRandom
getRandomR r = lift \$ getRandomR r

getRandom = lift getRandom
getRandomR r = lift \$ getRandomR r

instance (MonadState s m, RandomGen g) => MonadState s (RandomT g m) where
get = lift get
put s = lift \$ put s

local f m = RandomT \$ local f (unRT m)

instance (MonadWriter w m, RandomGen g, Monoid w) => MonadWriter w (RandomT g m) where
tell w = lift \$ tell w
listen m = RandomT \$ listen (unRT m)
pass m = RandomT \$ pass (unRT m)
```

You may also want a MonadRandom instance for IO:

```instance MonadRandom IO where
getRandom = randomIO
getRandomR = randomRIO
```

## Connection to stochastics

There is some 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 :: (MonadRandom m, Random a) => m 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

```  getRandomR (-10,10)
```

is a uniformly distributed random variable between -10 and 10, then

```  untilM (0/=) (getRandomR (-10,10))
```

is a random variable with a uniform distribution of $\{-10, \dots, -1, 1, \dots, 10\}$.