Îõ³h&Þ.      Safe-Inferred)*6×Ü9rvarA Promptà GADT describing a request for a primitive random variate. Random variable definitions will request their entropy via these prompts, and entropy sources will satisfy those requests. This data type is needed for creating  instance for rvar6An unsigned byte, uniformly distributed from 0 to 0xffrvar?An unsigned 16-bit word, uniformly distributed from 0 to 0xffffrvarÃAn unsigned 32-bit word, uniformly distributed from 0 to 0xffffffffrvarËAn unsigned 64-bit word, uniformly distributed from 0 to 0xffffffffffffffffrvarA uniformly distributed  of length n bytes Safe-Inferred )*ÁÂÃÄÅ×ÜþrvarèA 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: £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 xÜTo run the random walk it must first be initialized, after which it can be sampled as usual: Ìdo rw <- rwalkIO x <- sampleRVarT rw y <- sampleRVarT rw ...ÜThe 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 newËInvocation is straightforward (although a bit noisy) if you're used to MTL: Ùrwalk :: Int -> Double -> StdGen -> ([Double], StdGen) rwalk count start gen = flip evalState start . flip runStateT gen . sampleRVarTWith MTL.lift $ replicateM count rwalkState rvarëAn 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: ¹normalPair = 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:ËUsing an immutable pseudo-random number generator that has an instance for  with   monad:,import qualified Data.Random as Fu (uniform)import System.Random (mkStdGen)%import Control.Monad.State (runState)ÊrunState (sampleStateRVar (Fu.uniform 1 (100 :: Integer))) (mkStdGen 2021)Æ(79,StdGen {unStdGen = SMGen 4687568268719557181 4805600293067301895})ÇUsing a mutable pseud-random number generator that has an instance for ! with " monad.,import qualified Data.Random as Fu (uniform)!import System.Random.MWC (create)(import Control.Monad.Reader (runReaderT)+import qualified Data.Vector.Storable as VSÝinitialize (VS.singleton 2021) >>= runReaderT (sampleReaderRVar (uniform 1 (100 :: Integer)))8 rvarSample random variable using  generator as source of entropy rvar "Run" an  Ä - samples the random variable from the provided source of entropy. rvar sampleRVar x is equivalent to  runRVar x  StdRandom.rvar "Runs" an *, sampling the random variable it defines.ùThe 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)ÆOne example of a useful non-standard lifting would be one that takes State s 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  Ç 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  às would have to use higher-ranked types to enforce the same kind of isolation and polymorphism.rvarsampleRVarTWith lift x is equivalent to runRVarTWith lift x  StdRandom.rvarsampleRVarTWith lift x is equivalent to runRVarTWith lift x  StdRandom.  #       !"#$%&'()*+,'(*-./#rvar-0.3.0.2-BurKtbHfVAd77GYTsNgigK Data.RVarData.RVar.PrimSystem.Random.Stateful StatefulGenRVarTPrim PrimWord8 PrimWord16 PrimWord32 PrimWord64PrimShortByteStringRGenRVarpureRVarrunRVarsampleReaderRVarsampleStateRVarrunRVarTsampleStateRVarTsampleReaderRVarT runRVarTWithsampleReaderRVarTWithsampleStateRVarTWith uniformRVarTuniformRangeRVarT$fMonadIORVarT$fMonadTransRVarT$fMonadPromptPrimRVarT$fApplicativeRVarT $fMonadRVarT$fFunctorRVarT$fStatefulGenRGenRVarTbytestring-0.11.3.1Data.ByteString.Short.InternalShortByteStringghc-prim GHC.TypesIO%random-1.2.1.1-Gq45Dc4CYFB1uLVOCO4LJ3System.Random.Internal RandomGentransformers-0.5.6.2Control.Monad.Trans.State.LazyStateTControl.Monad.Trans.ReaderReaderT