```{-# LANGUAGE DeriveDataTypeable #-}
-- |
-- Module    : Statistics.Distribution.FDistribution
-- Copyright : (c) 2011 Aleksey Khudyakov
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Fisher F distribution
module Statistics.Distribution.FDistribution (
FDistribution
, fDistribution
, fDistributionNDF1
, fDistributionNDF2
) where

import qualified Statistics.Distribution as D
import Data.Typeable         (Typeable)
import Numeric.SpecFunctions (logBeta, incompleteBeta, invIncompleteBeta)

-- | Student-T distribution
data FDistribution = F { fDistributionNDF1 :: {-# UNPACK #-} !Double
, fDistributionNDF2 :: {-# UNPACK #-} !Double
, _pdfFactor        :: {-# UNPACK #-} !Double
}

fDistribution :: Int -> Int -> FDistribution
fDistribution n m
| n > 0 && m > 0 =
let n' = fromIntegral n
m' = fromIntegral m
f' = 0.5 * (log m' * m' + log n' * n') - logBeta (0.5*n') (0.5*m')
in F n' m' f'
| otherwise =
error "Statistics.Distribution.FDistribution.fDistribution: non-positive number of degrees of freedom"

instance D.Distribution FDistribution where
cumulative = cumulative

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

cumulative :: FDistribution -> Double -> Double
cumulative (F n m _) x
| x > 0     = let y = n*x in incompleteBeta (0.5 * n) (0.5 * m) (y / (m + y))
| otherwise = 0

density :: FDistribution -> Double -> Double
density (F n m fac) x
| x > 0     = exp \$ fac + log x * (0.5 * n - 1) - log(m + n*x) * 0.5 * (n + m)
| otherwise = 0

quantile :: FDistribution -> Double -> Double
quantile (F n m _) p
| p >= 0 && p <= 1 =
let x = invIncompleteBeta (0.5 * n) (0.5 * m) p
in m * x / (n * (1 - x))
| otherwise =
error \$ "Statistics.Distribution.Uniform.quantile: p must be in [0,1] range. Got: "++show p

instance D.MaybeMean FDistribution where
maybeMean (F _ m _) | m > 2     = Just \$ m / (m - 2)
| otherwise = Nothing

instance D.MaybeVariance FDistribution where
maybeStdDev (F n m _)
| m > 4     = Just \$ 2 * sqr m * (m + n - 2) / (n * sqr (m - 2) * (m - 4))
| otherwise = Nothing

sqr :: Double -> Double
sqr x = x * x
{-# INLINE sqr #-}
```