    {-# LANGUAGE DeriveDataTypeable #-}
    -- |
    -- Module    : Statistics.Distribution.Normal
    -- Copyright : (c) 2009 Bryan O'Sullivan
    -- License   : BSD3
    --
    -- Maintainer  : bos@serpentine.com
    -- Stability   : experimental
    -- Portability : portable
    --
    -- The normal distribution.  This is a continuous probability
    -- distribution that describes data that cluster around a mean.

    module Statistics.Distribution.Normal
        (
          NormalDistribution
        -- * Constructors
        , normalDistr
        , normalFromSample
        , standard
        ) where

    import Data.Number.Erf (erfc)
    import Data.Typeable (Typeable)
    import Numeric.MathFunctions.Constants (m_sqrt_2, m_sqrt_2_pi)
    import qualified Statistics.Distribution as D
    import qualified Statistics.Sample as S
    import qualified System.Random.MWC.Distributions as MWC

    -- | The normal distribution.
    data NormalDistribution = ND {
          mean       :: {-# UNPACK #-} !Double
        , stdDev     :: {-# UNPACK #-} !Double
        , ndPdfDenom :: {-# UNPACK #-} !Double
        , ndCdfDenom :: {-# UNPACK #-} !Double
        } deriving (Eq, Read, Show, Typeable)

    instance D.Distribution NormalDistribution where
        cumulative      = cumulative
        complCumulative = complCumulative

    instance D.ContDistr NormalDistribution where
        density    = density
        quantile   = quantile

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

    instance D.Mean NormalDistribution where
        mean = mean

    instance D.MaybeVariance NormalDistribution where
        maybeStdDev   = Just . D.stdDev
        maybeVariance = Just . D.variance

    instance D.Variance NormalDistribution where
        stdDev = stdDev

    instance D.ContGen NormalDistribution where
        genContVar d = MWC.normal (mean d) (stdDev d)
        {-# INLINE genContVar #-}

    -- | Standard normal distribution with mean equal to 0 and variance equal to 1
    standard :: NormalDistribution
    standard = ND { mean       = 0.0
                  , stdDev     = 1.0
                  , ndPdfDenom = m_sqrt_2_pi
                  , ndCdfDenom = m_sqrt_2
                  }

    -- | Create normal distribution from parameters.
    --
    -- IMPORTANT: prior to 0.10 release second parameter was variance not
    -- standard deviation.
    normalDistr :: Double            -- ^ Mean of distribution
                -> Double            -- ^ Standard deviation of distribution
                -> NormalDistribution
    normalDistr m sd
      | sd > 0    = ND { mean       = m
                       , stdDev     = sd
                       , ndPdfDenom = m_sqrt_2_pi * sd
                       , ndCdfDenom = m_sqrt_2 * sd
                       }
      | otherwise =
        error $ "Statistics.Distribution.Normal.normalDistr: standard deviation must be positive. Got " ++ show sd

    -- | Create distribution using parameters estimated from
    --   sample. Variance is estimated using maximum likelihood method
    --   (biased estimation).
    normalFromSample :: S.Sample -> NormalDistribution
    normalFromSample a = normalDistr (S.mean a) (S.stdDev a)

    density :: NormalDistribution -> Double -> Double
    density d x = exp (-xm * xm / (2 * sd * sd)) / ndPdfDenom d
        where xm = x - mean d
              sd = stdDev d

    cumulative :: NormalDistribution -> Double -> Double
    cumulative d x = erfc ((mean d - x) / ndCdfDenom d) / 2

    complCumulative :: NormalDistribution -> Double -> Double
    complCumulative d x = erfc ((x - mean d) / ndCdfDenom d) / 2

    quantile :: NormalDistribution -> Double -> Double
    quantile d p
      | p == 0         = -inf
      | p == 1         = inf
      | p == 0.5       = mean d
      | p > 0 && p < 1 = x * stdDev d + mean d
      | otherwise      =
        error $ "Statistics.Distribution.Normal.quantile: p must be in [0,1] range. Got: "++show p
      where x          = D.findRoot standard p 0 (-100) 100
            inf        = 1/0