```{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module    : Statistics.Distribution.StudentT
-- Copyright : (c) 2011 Aleksey Khudyakov
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Student-T distribution
module Statistics.Distribution.StudentT (
StudentT
, studentT
, studentTndf
, studentTUnstandardized
) where

import Data.Binary (Binary)
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import qualified Statistics.Distribution as D
import Statistics.Distribution.Transform (LinearTransform (..))
import Numeric.SpecFunctions (logBeta, incompleteBeta, invIncompleteBeta)

-- | Student-T distribution
newtype StudentT = StudentT { studentTndf :: Double }
deriving (Eq, Show, Read, Typeable, Data, Generic)

instance Binary StudentT

-- | Create Student-T distribution. Number of parameters must be positive.
studentT :: Double -> StudentT
studentT ndf
| ndf > 0   = StudentT ndf
| otherwise = modErr "studentT" "non-positive number of degrees of freedom"

instance D.Distribution StudentT where
cumulative = cumulative

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

cumulative :: StudentT -> Double -> Double
cumulative (StudentT ndf) x
| x > 0     = 1 - 0.5 * ibeta
| otherwise = 0.5 * ibeta
where
ibeta = incompleteBeta (0.5 * ndf) 0.5 (ndf / (ndf + x*x))

density :: StudentT -> Double -> Double
density (StudentT ndf) x =
exp( log (ndf / (ndf + x*x)) * (0.5 * (1 + ndf)) - logBeta 0.5 (0.5 * ndf) ) / sqrt ndf

quantile :: StudentT -> Double -> Double
quantile (StudentT ndf) p
| p >= 0 && p <= 1 =
let x = invIncompleteBeta (0.5 * ndf) 0.5 (2 * min p (1 - p))
in case sqrt \$ ndf * (1 - x) / x of
r | p < 0.5   -> -r
| otherwise -> r
| otherwise = modErr "quantile" \$ "p must be in [0,1] range. Got: "++show p

instance D.MaybeMean StudentT where
maybeMean (StudentT ndf) | ndf > 1   = Just 0
| otherwise = Nothing

instance D.MaybeVariance StudentT where
maybeVariance (StudentT ndf) | ndf > 2   = Just \$! ndf / (ndf - 2)
| otherwise = Nothing

instance D.ContGen StudentT where
genContVar = D.genContinous

-- | Create an unstandardized Student-t distribution.
studentTUnstandardized :: Double -- ^ Number of degrees of freedom
-> Double -- ^ Central value (0 for standard Student T distribution)
-> Double -- ^ Scale parameter
-> LinearTransform StudentT
studentTUnstandardized ndf mu sigma
| sigma > 0 = LinearTransform mu sigma \$ studentT ndf
| otherwise = modErr "studentTUnstandardized" \$ "sigma must be > 0. Got: " ++ show sigma

modErr :: String -> String -> a
modErr fun msg = error \$ "Statistics.Distribution.StudentT." ++ fun ++ ": " ++ msg
```