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

module Statistics.Distribution
(
Distribution(..)
, Mean(..)
, Variance(..)
, findRoot
) where

-- | The interface shared by all probability distributions.
class Distribution d where
-- | Probability density function. The probability that a
-- stochastic variable /x/ has the value /X/, i.e. P(/x/=/X/).
probability :: d -> Double -> Double

-- | Cumulative distribution function.  The probability that a
-- stochastic variable /x/ is less than /X/, i.e. P(/x/</X/).
cumulative  :: d -> Double -> Double

-- | Inverse of the cumulative distribution function.  The value
-- /X/ for which P(/x/</X/).
inverse     :: d -> Double -> Double

class Distribution d => Mean d where
mean :: d -> Double

class Mean d => Variance d where
variance :: d -> 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 :: Distribution d => d
-> 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
(lo',hi') | err < 0   = (x, hi)
| otherwise = (lo, x)
pdf                   = probability d x
(dx',x') | pdf /= 0   = (err / pdf, x - dx)
| otherwise  = (dx, x)
(dx'',x'')
| x' < lo' || x' > hi' || pdf == 0 = (x'-x, (lo + hi) / 2)
| otherwise                        = (dx',  x')
accuracy = 1e-15
maxIters = 150
```