module Statistics.Distribution.Normal
(
NormalDistribution
, normalDistr
, normalFromSample
, standard
) where
import Control.Exception (assert)
import Data.Number.Erf (erfc)
import Data.Typeable (Typeable)
import Statistics.Constants (m_sqrt_2, m_sqrt_2_pi)
import qualified Statistics.Distribution as D
import qualified Statistics.Sample as S
data NormalDistribution = ND {
mean :: !Double
, variance :: !Double
, ndPdfDenom :: !Double
, ndCdfDenom :: !Double
} deriving (Eq, Read, Show, Typeable)
instance D.Distribution NormalDistribution where
cumulative = cumulative
instance D.ContDistr NormalDistribution where
density = density
quantile = quantile
instance D.Variance NormalDistribution where
variance = variance
instance D.Mean NormalDistribution where
mean = mean
standard :: NormalDistribution
standard = ND { mean = 0.0
, variance = 1.0
, ndPdfDenom = m_sqrt_2_pi
, ndCdfDenom = m_sqrt_2
}
normalDistr :: Double
-> Double
-> NormalDistribution
normalDistr m v = assert (v > 0)
ND { mean = m
, variance = v
, ndPdfDenom = m_sqrt_2_pi * sv
, ndCdfDenom = m_sqrt_2 * sv
}
where sv = sqrt v
normalFromSample :: S.Sample -> NormalDistribution
normalFromSample a = normalDistr (S.mean a) (S.variance a)
density :: NormalDistribution -> Double -> Double
density d x = exp (xm * xm / (2 * variance d)) / ndPdfDenom d
where xm = x mean d
cumulative :: NormalDistribution -> Double -> Double
cumulative d x = erfc ((mean d x) / ndCdfDenom d) / 2
quantile :: NormalDistribution -> Double -> Double
quantile d p
| p < 0 || p > 1 = inf/inf
| p == 0 = inf
| p == 1 = inf
| p == 0.5 = mean d
| otherwise = x * sqrt (variance d) + mean d
where x = D.findRoot standard p 0 (100) 100
inf = 1/0