```{-# LANGUAGE DeriveDataTypeable #-}
-- |
-- Module    : Statistics.Distribution.Exponential
-- Copyright : (c) 2009 Bryan O'Sullivan
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- The exponential distribution.  This is the continunous probability
-- distribution of the times between events in a poisson process, in
-- which events occur continuously and independently at a constant
-- average rate.

module Statistics.Distribution.Exponential
(
ExponentialDistribution
-- * Constructors
, exponential
, exponentialFromSample
-- * Accessors
, edLambda
) where

import Data.Typeable (Typeable)
import qualified Statistics.Distribution as D
import qualified Statistics.Sample as S
import Statistics.Types (Sample)

newtype ExponentialDistribution = ED {
edLambda :: Double
} deriving (Eq, Read, Show, Typeable)

instance D.Distribution ExponentialDistribution where
cumulative      = cumulative
complCumulative = complCumulative

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

instance D.Mean ExponentialDistribution where
mean (ED l) = 1 / l
{-# INLINE mean #-}

instance D.Variance ExponentialDistribution where
variance (ED l) = 1 / (l * l)
{-# INLINE variance #-}

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

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

cumulative :: ExponentialDistribution -> Double -> Double
cumulative (ED l) x | x <= 0    = 0
| otherwise = 1 - exp (-l * x)
{-# INLINE cumulative #-}

complCumulative :: ExponentialDistribution -> Double -> Double
complCumulative (ED l) x | x <= 0    = 1
| otherwise = exp (-l * x)
{-# INLINE complCumulative #-}

density :: ExponentialDistribution -> Double -> Double
density (ED l) x | x < 0     = 0
| otherwise = l * exp (-l * x)
{-# INLINE density #-}

quantile :: ExponentialDistribution -> Double -> Double
quantile (ED l) p
| p == 1          = 1 / 0
| p >= 0 && p < 1 = -log (1 - p) / l
| otherwise       =
error \$ "Statistics.Distribution.Exponential.quantile: p must be in [0,1] range. Got: "++show p
{-# INLINE quantile #-}

-- | Create an exponential distribution.
exponential :: Double            -- ^ &#955; (scale) parameter.
-> ExponentialDistribution
exponential l
| l <= 0 =
error \$ "Statistics.Distribution.Exponential.exponential: scale parameter must be positive. Got " ++ show l
| otherwise = ED l
{-# INLINE exponential #-}

-- | Create exponential distribution from sample. No tests are made to
-- check whether it truly is exponential.
exponentialFromSample :: Sample -> ExponentialDistribution
exponentialFromSample = ED . S.mean
{-# INLINE exponentialFromSample #-}
```