    {-# LANGUAGE DeriveDataTypeable #-}
    -- |
    -- Module    : Statistics.Distribution.Poisson
    -- Copyright : (c) 2009, 2011 Bryan O'Sullivan
    -- License   : BSD3
    --
    -- Maintainer  : bos@serpentine.com
    -- Stability   : experimental
    -- Portability : portable
    --
    -- The Poisson distribution.  This is the discrete probability
    -- distribution of a number of events occurring in a fixed interval if
    -- these events occur with a known average rate, and occur
    -- independently from each other within that interval.

    module Statistics.Distribution.Poisson
        (
          PoissonDistribution
        -- * Constructors
        , poisson
        -- * Accessors
        , poissonLambda
        -- * References
        -- $references
        ) where

    import Data.Typeable (Typeable)
    import qualified Statistics.Distribution as D
    import qualified Statistics.Distribution.Poisson.Internal as I
    import Numeric.SpecFunctions (incompleteGamma)

    newtype PoissonDistribution = PD {
          poissonLambda :: Double
        } deriving (Eq, Read, Show, Typeable)

    instance D.Distribution PoissonDistribution where
        cumulative (PD lambda) x
          | x < 0     = 0
          | otherwise = 1 - incompleteGamma (fromIntegral (floor x + 1 :: Int)) lambda
        {-# INLINE cumulative #-}

    instance D.DiscreteDistr PoissonDistribution where
        probability (PD lambda) x = I.probability lambda (fromIntegral x)
        {-# INLINE probability #-}

    instance D.Variance PoissonDistribution where
        variance = poissonLambda
        {-# INLINE variance #-}

    instance D.Mean PoissonDistribution where
        mean = poissonLambda
        {-# INLINE mean #-}

    instance D.MaybeMean PoissonDistribution where
        maybeMean = Just . D.mean

    instance D.MaybeVariance PoissonDistribution where
        maybeStdDev   = Just . D.stdDev

    -- | Create Poisson distribution.
    poisson :: Double -> PoissonDistribution
    poisson l
      | l >=  0   = PD l
      | otherwise = error $ "Statistics.Distribution.Poisson.poisson:\
                            \ lambda must be non-negative. Got " ++ show l
    {-# INLINE poisson #-}

    -- $references
    --
    -- * Loader, C. (2000) Fast and Accurate Computation of Binomial
    --   Probabilities. <http://projects.scipy.org/scipy/raw-attachment/ticket/620/loader2000Fast.pdf>