```{-# LANGUAGE DeriveDataTypeable #-}
-- |
-- Module    : Statistics.Distribution.Normal
-- Copyright : (c) 2009 Bryan O'Sullivan
--
-- 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
, fromParams
, fromSample
, standard
) where

import Control.Exception (assert)
import Data.Number.Erf (erfc)
import Data.Typeable (Typeable)
import Statistics.Constants (m_huge, m_sqrt_2, m_sqrt_2_pi)
import Statistics.Types (Sample)
import qualified Statistics.Distribution as D
import qualified Statistics.Sample as S

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

instance D.Distribution NormalDistribution where
probability = probability
cumulative  = cumulative
inverse     = inverse

instance D.Variance NormalDistribution where
variance = variance

instance D.Mean NormalDistribution where
mean = mean

standard :: NormalDistribution
standard = ND {
mean = 0.0
, variance = 1.0
, ndPdfDenom = m_sqrt_2_pi
, ndCdfDenom = m_sqrt_2
}

fromParams :: Double -> Double -> NormalDistribution
fromParams m v = assert (v > 0)
ND {
mean = m
, variance = v
, ndPdfDenom = m_sqrt_2_pi * sv
, ndCdfDenom = m_sqrt_2 * sv
}
where sv = sqrt v

fromSample :: Sample -> NormalDistribution
fromSample a = fromParams (S.mean a) (S.variance a)

probability :: NormalDistribution -> Double -> Double
probability d x = exp (-xm * xm / (2 * variance d)) / ndPdfDenom d
where xm = x - mean d

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

inverse :: NormalDistribution -> Double -> Double
inverse d p
| p == 0    = -m_huge
| p == 1    = m_huge
| p == 0.5  = mean d
| otherwise = x * sqrt (variance d) + mean d
where x     = D.findRoot standard p 0 (-100) 100
```