```{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- 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.Binary (Binary)
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import qualified Statistics.Distribution as D
import qualified Statistics.Distribution.Poisson.Internal as I
import Numeric.SpecFunctions (incompleteGamma,logFactorial)
import Numeric.MathFunctions.Constants (m_neg_inf)
import Data.Binary (put, get)

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

instance Binary PoissonDistribution where
get = fmap PD get
put = put . poissonLambda

instance D.Distribution PoissonDistribution where
cumulative (PD lambda) x
| x < 0        = 0
| isInfinite x = 1
| isNaN      x = error "Statistics.Distribution.Poisson.cumulative: NaN input"
| 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)
logProbability (PD lambda) i
| i < 0     = m_neg_inf
| otherwise = log lambda * fromIntegral i - logFactorial i - lambda
{-# 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

instance D.Entropy PoissonDistribution where
entropy (PD lambda) = I.poissonEntropy lambda

instance D.MaybeEntropy PoissonDistribution where
maybeEntropy = Just . D.entropy

-- | 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>
-- * Adell, J., Lekuona, A., and Yu, Y. (2010) Sharp Bounds on the
--   Entropy of the Poisson Law and Related Quantities
--   <http://arxiv.org/pdf/1001.2897.pdf>
```