úÎ9k8     None !"246HMèA random variable with access to operations in an underlying monad. Useful examples include any form of state for implementing random processes with hysteresis, or writer monads for implementing tracing of complicated algorithms.;For example, a simple random walk can be implemented as an   value: ÿ;rwalkIO :: IO (RVarT IO Double) rwalkIO d = do lastVal <- newIORef 0 let x = do prev <- lift (readIORef lastVal) change <- rvarT StdNormal let new = prev + change lift (writeIORef lastVal new) return new return x\To run the random walk it must first be initialized, after which it can be sampled as usual: Ldo rw <- rwalkIO x <- sampleRVarT rw y <- sampleRVarT rw ...\The same random-walk process as above can be implemented using MTL types as follows (using !import Control.Monad.Trans as MTL): ¸rwalkState :: RVarT (State Double) Double rwalkState = do prev <- MTL.lift get change <- rvarT StdNormal let new = prev + change MTL.lift (put new) return newKInvocation is straightforward (although a bit noisy) if you're used to MTL: Úrwalk :: Int -> Double -> StdGen -> ([Double], StdGen) rwalk count start gen = flip evalState start . flip runStateT gen . sampleRVarTWith MTL.lift $ replicateM count rwalkState lAn opaque type modeling a "random variable" - a value which depends on the outcome of some random event.  ?s can be conveniently defined by an imperative-looking style: ÁnormalPair = do u <- stdUniform t <- stdUniform let r = sqrt (-2 * log u) theta = (2 * pi) * t x = r * cos theta y = r * sin theta return (x,y)OR by a more applicative style: logNormal = exp <$> stdNormal>Once defined (in any style), there are several ways to sample  s:In a monad, using a : +runRVar (uniform 1 100) DevRandom :: IO IntIn a monad, using a  instance: .sampleRVar (uniform 1 100) :: State PureMT Int1As a pure function transforming a functional RNG: 6sampleState (uniform 1 100) :: StdGen -> (Int, StdGen)(where #sampleState = runState . sampleRVar)  "Run" an  D - samples the random variable from the provided source of entropy.  sampleRVar x is equivalent to  runRVar x  StdRandom. "Runs" an *, sampling the random variable it defines.zThe first argument lifts the base monad into the sampling monad. This operation must obey the "monad transformer" laws: ?lift . return = return lift (x >>= f) = (lift x) >>= (lift . f)FOne example of a useful non-standard lifting would be one that takes State sB to another monad with a different state representation (such as IO with the state mapped to an IORef): ªembedState :: (Monad m) => m s -> (s -> m ()) -> State s a -> m a embedState get put = \m -> do s <- get (res,s) <- return (runState m s) put s return res:The ability to lift is very important - without it, every  È would have to either be given access to the full capability of the monad in which it will eventually be sampled (which, incidentally, would also have to be monomorphic so you couldn't sample one  4 in more than one monad) or functions manipulating  as would have to use higher-ranked types to enforce the same kind of isolation and polymorphism.sampleRVarTWith lift x is equivalent to runRVarTWith lift x  StdRandom.            rvar-0.2.0.2 Data.RVarrandom-source-0.3.0.6Data.Random.Internal.SourcegetRandomNByteIntegergetRandomDoublegetRandomWord64getRandomWord32getRandomWord16getRandomWord8 MonadRandom RandomSourceRVarTRVarrunRVar sampleRVarrunRVarT sampleRVarT runRVarTWithsampleRVarTWithghc-prim GHC.TypesIOunRVarT$fMonadIORVarT$fMonadTransRVarT$fMonadPromptPrimRVarT$fApplicativeRVarT$fMonadRandomRVarT $fMonadRVarT$fFunctorRVarT