```{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module    : Statistics.Distribution.Gamma
-- Copyright : (c) 2009, 2011 Bryan O'Sullivan
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- The gamma distribution.  This is a continuous probability
-- distribution with two parameters, /k/ and &#977;. If /k/ is
-- integral, the distribution represents the sum of /k/ independent
-- exponentially distributed random variables, each of which has a
-- mean of &#977;.

module Statistics.Distribution.Gamma
(
-- * Constructors
-- * Accessors
, gdShape
, gdScale
) where

import Data.Aeson (FromJSON, ToJSON)
import Control.Applicative ((<\$>), (<*>))
import Data.Binary (Binary)
import Data.Binary (put, get)
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import Numeric.MathFunctions.Constants (m_pos_inf, m_NaN, m_neg_inf)
import Numeric.SpecFunctions (incompleteGamma, invIncompleteGamma, logGamma, digamma)
import Statistics.Distribution.Poisson.Internal as Poisson
import qualified Statistics.Distribution as D
import qualified System.Random.MWC.Distributions as MWC

-- | The gamma distribution.
gdShape :: {-# UNPACK #-} !Double -- ^ Shape parameter, /k/.
, gdScale :: {-# UNPACK #-} !Double -- ^ Scale parameter, &#977;.
} deriving (Eq, Read, Show, Typeable, Data, Generic)

put (GD x y) = put x >> put y
get = GD <\$> get <*> get

-- | Create gamma distribution. Both shape and scale parameters must
-- be positive.
gammaDistr :: Double            -- ^ Shape parameter. /k/
-> Double            -- ^ Scale parameter, &#977;.
| k     <= 0 = error \$ msg ++ "shape must be positive. Got " ++ show k
| theta <= 0 = error \$ msg ++ "scale must be positive. Got " ++ show theta
| otherwise  = improperGammaDistr k theta

-- | Create gamma distribution. This constructor do not check whether
--   parameters are valid
improperGammaDistr :: Double            -- ^ Shape parameter. /k/
-> Double            -- ^ Scale parameter, &#977;.

cumulative = cumulative

density    = density
logDensity (GD k theta) x
| x <= 0    = m_neg_inf
| otherwise = log x * (k - 1) - (x / theta) - logGamma k - log theta * k
quantile   = quantile

variance (GD a l) = a * l * l

mean (GD a l) = a * l

maybeMean = Just . D.mean

maybeStdDev   = Just . D.stdDev
maybeVariance = Just . D.variance

maybeEntropy (GD a l)
| a > 0 && l > 0 =
Just \$
a
+ log l
+ logGamma a
+ (1-a) * digamma a
| otherwise = Nothing

genContVar (GD a l) = MWC.gamma a l

density :: GammaDistribution -> Double -> Double
density (GD a l) x
| a < 0 || l <= 0   = m_NaN
| x <= 0            = 0
| a == 0            = if x == 0 then m_pos_inf else 0
| x == 0            = if a < 1 then m_pos_inf else if a > 1 then 0 else 1/l
| a < 1             = Poisson.probability (x/l) a * a / x
| otherwise         = Poisson.probability (x/l) (a-1) / l

cumulative :: GammaDistribution -> Double -> Double
cumulative (GD k l) x
| x <= 0    = 0
| otherwise = incompleteGamma k (x/l)

quantile :: GammaDistribution -> Double -> Double
quantile (GD k l) p
| p == 0         = 0
| p == 1         = 1/0
| p > 0 && p < 1 = l * invIncompleteGamma k p
| otherwise      =
error \$ "Statistics.Distribution.Gamma.quantile: p must be in [0,1] range. Got: "++show p
```