```{-# LANGUAGE BangPatterns, ScopedTypeVariables #-}
-- |
-- Module    : Statistics.Distribution
-- Copyright : (c) 2009 Bryan O'Sullivan
-- License   : BSD3
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Types classes for probability distrubutions

module Statistics.Distribution
(
-- * Type classes
Distribution(..)
, DiscreteDistr(..)
, ContDistr(..)
, Mean(..)
, Variance(..)
-- * Helper functions
, findRoot
, sumProbabilities
) where

import qualified Data.Vector.Unboxed as U

-- | Type class common to all distributions. Only c.d.f. could be
-- defined for both discrete and continous distributions.
class Distribution d where
-- | Cumulative distribution function.  The probability that a
-- random variable /X/ is less or equal than /x/,
-- i.e. P(/X/&#8804;/x/).
cumulative :: d -> Double -> Double

-- | Discrete probability distribution.
class Distribution  d => DiscreteDistr d where
-- | Probability of n-th outcome.
probability :: d -> Int -> Double

-- | Continuous probability distributuion
class Distribution d => ContDistr d where
-- | Probability density function. Probability that random
-- variable /X/ lies in the infinitesimal interval
-- [/x/,/x+/&#948;/x/) equal to /density(x)/&#8901;&#948;/x/
density :: d -> Double -> Double

-- | Inverse of the cumulative distribution function. The value
-- /x/ for which P(/X/&#8804;/x/) = /p/.
quantile :: d -> Double -> Double

-- | Type class for distributions with mean.
class Distribution d => Mean d where
mean :: d -> Double

-- | Type class for distributions with variance.
class Mean d => Variance d where
variance :: d -> Double

data P = P {-# UNPACK #-} !Double {-# UNPACK #-} !Double

-- | Approximate the value of /X/ for which P(/x/>/X/)=/p/.
--
-- This method uses a combination of Newton-Raphson iteration and
-- bisection with the given guess as a starting point.  The upper and
-- lower bounds specify the interval in which the probability
-- distribution reaches the value /p/.
findRoot :: ContDistr d =>
d                   -- ^ Distribution
-> Double              -- ^ Probability /p/
-> Double              -- ^ Initial guess
-> Double              -- ^ Lower bound on interval
-> Double              -- ^ Upper bound on interval
-> Double
findRoot d prob = loop 0 1
where
loop !(i::Int) !dx !x !lo !hi
| abs dx <= accuracy || i >= maxIters = x
| otherwise                           = loop (i+1) dx'' x'' lo' hi'
where
err                   = cumulative d x - prob
P lo' hi' | err < 0   = P x hi
| otherwise = P lo x
pdf                   = density d x
P dx' x' | pdf /= 0   = P (err / pdf) (x - dx)
| otherwise  = P dx x
P dx'' x''
| x' < lo' || x' > hi' || pdf == 0 = let y = (lo' + hi') / 2
in  P (y-x) y
| otherwise                        = P dx' x'
accuracy = 1e-15
maxIters = 150

-- | Sum probabilities in inclusive interval.
sumProbabilities :: DiscreteDistr d => d -> Int -> Int -> Double
sumProbabilities d low hi =
-- Return value is forced to be less than 1 to guard againist roundoff errors.
-- ATTENTION! this check should be removed for testing or it could mask bugs.
min 1 . U.sum . U.map (probability d) \$ U.enumFromTo low hi
{-# INLINE sumProbabilities #-}
```