```{-# LANGUAGE DeriveDataTypeable #-}
-- |
-- Module    : Statistics.Distribution.ChiSquared
-- Copyright : (c) 2010 Alexey Khudyakov
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- The chi-squared distribution. This is a continuous probability
-- distribution of sum of squares of k independent standard normal
-- distributions. It's commonly used in statistical tests
module Statistics.Distribution.ChiSquared (
ChiSquared
-- Constructors
, chiSquared
, chiSquaredNDF
) where

import Data.Typeable        (Typeable)
import Statistics.Math      (incompleteGamma,invIncompleteGamma,logGamma)

import qualified Statistics.Distribution as D

-- | Chi-squared distribution
newtype ChiSquared = ChiSquared Int
deriving (Show,Typeable)

-- | Get number of degrees of freedom
chiSquaredNDF :: ChiSquared -> Int
chiSquaredNDF (ChiSquared ndf) = ndf
{-# INLINE chiSquaredNDF #-}

-- | Construct chi-squared distribution. Number of degrees of freedom
--   must be positive.
chiSquared :: Int -> ChiSquared
chiSquared n
| n <= 0    = error \$
"Statistics.Distribution.ChiSquared.chiSquared: N.D.F. must be positive. Got " ++ show n
| otherwise = ChiSquared n
{-# INLINE chiSquared #-}

instance D.Distribution ChiSquared where
cumulative = cumulative

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

instance D.Mean ChiSquared where
mean (ChiSquared ndf) = fromIntegral ndf
{-# INLINE mean #-}

instance D.Variance ChiSquared where
variance (ChiSquared ndf) = fromIntegral (2*ndf)
{-# INLINE variance #-}

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

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

cumulative :: ChiSquared -> Double -> Double
cumulative chi x
| x <= 0    = 0
| otherwise = incompleteGamma (ndf/2) (x/2)
where
ndf = fromIntegral \$ chiSquaredNDF chi
{-# INLINE cumulative #-}

density :: ChiSquared -> Double -> Double
density chi x
| x <= 0    = 0
| otherwise = exp \$ log x * (ndf2 - 1) - x2 - logGamma ndf2 - log 2 * ndf2
where
ndf  = fromIntegral \$ chiSquaredNDF chi
ndf2 = ndf/2
x2   = x/2
{-# INLINE density #-}

quantile :: ChiSquared -> Double -> Double
quantile (ChiSquared ndf) p
| p == 0         = 0
| p == 1         = 1/0
| p > 0 && p < 1 = 2 * invIncompleteGamma (fromIntegral ndf / 2) p
| otherwise      =
error \$ "Statistics.Distribution.ChiSquared.quantile: p must be in [0,1] range. Got: "++show p
{-# INLINE quantile #-}
```