{-# LANGUAGE OverloadedStrings #-}{-# LANGUAGE PatternGuards #-}{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}-- |-- Module : Statistics.Distribution.Binomial-- Copyright : (c) 2009 Bryan O'Sullivan-- License : BSD3---- Maintainer : bos@serpentine.com-- Stability : experimental-- Portability : portable---- The binomial distribution. This is the discrete probability-- distribution of the number of successes in a sequence of /n/-- independent yes\/no experiments, each of which yields success with-- probability /p/.moduleStatistics.Distribution.Binomial(BinomialDistribution -- * Constructors,binomial ,binomialE -- * Accessors,bdTrials ,bdProbability )whereimportControl.ApplicativeimportData.Aeson(FromJSON(..),ToJSON,Value(..),(.:))importData.Binary(Binary(..))importData.Data(Data,Typeable)importGHC.Generics(Generic)importNumeric.SpecFunctions(choose,logChoose,incompleteBeta,log1p)importNumeric.MathFunctions.Constants(m_epsilon,m_tiny)importqualifiedStatistics.Distribution asDimportqualifiedStatistics.Distribution.Poisson.Internal asIimportStatistics.Internal -- | The binomial distribution.dataBinomialDistribution =BD {BinomialDistribution -> Int
bdTrials ::{-# UNPACK#-}!Int-- ^ Number of trials.,BinomialDistribution -> Double
bdProbability ::{-# UNPACK#-}!Double-- ^ Probability.}deriving(BinomialDistribution -> BinomialDistribution -> Bool
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-> (forall x. Rep BinomialDistribution x -> BinomialDistribution)
-> Generic BinomialDistribution
forall x. Rep BinomialDistribution x -> BinomialDistribution
forall x. BinomialDistribution -> Rep BinomialDistribution x
forall a.
(forall x. a -> Rep a x) -> (forall x. Rep a x -> a) -> Generic a
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to :: forall x. Rep BinomialDistribution x -> BinomialDistribution
Generic)instanceShowBinomialDistribution whereshowsPrec :: Int -> BinomialDistribution -> ShowS
showsPrec Int
i (BD Int
n Double
p )=[Char] -> Int -> Double -> Int -> ShowS
forall a b. (Show a, Show b) => [Char] -> a -> b -> Int -> ShowS
defaultShow2 [Char]
"binomial"Int
n Double
p Int
i instanceReadBinomialDistribution wherereadPrec :: ReadPrec BinomialDistribution
readPrec =[Char]
-> (Int -> Double -> Maybe BinomialDistribution)
-> ReadPrec BinomialDistribution
forall a b r.
(Read a, Read b) =>
[Char] -> (a -> b -> Maybe r) -> ReadPrec r
defaultReadPrecM2 [Char]
"binomial"Int -> Double -> Maybe BinomialDistribution
binomialE instanceToJSONBinomialDistribution instanceFromJSONBinomialDistribution whereparseJSON :: Value -> Parser BinomialDistribution
parseJSON (ObjectObject
v )=doInt
n <-Object
v Object -> Key -> Parser Int
forall a. FromJSON a => Object -> Key -> Parser a
.:Key
"bdTrials"Double
p <-Object
v Object -> Key -> Parser Double
forall a. FromJSON a => Object -> Key -> Parser a
.:Key
"bdProbability"Parser BinomialDistribution
-> (BinomialDistribution -> Parser BinomialDistribution)
-> Maybe BinomialDistribution
-> Parser BinomialDistribution
forall b a. b -> (a -> b) -> Maybe a -> b
maybe([Char] -> Parser BinomialDistribution
forall a. [Char] -> Parser a
forall (m :: * -> *) a. MonadFail m => [Char] -> m a
fail([Char] -> Parser BinomialDistribution)
-> [Char] -> Parser BinomialDistribution
forall a b. (a -> b) -> a -> b
$Int -> Double -> [Char]
errMsg Int
n Double
p )BinomialDistribution -> Parser BinomialDistribution
forall a. a -> Parser a
forall (m :: * -> *) a. Monad m => a -> m a
return(Maybe BinomialDistribution -> Parser BinomialDistribution)
-> Maybe BinomialDistribution -> Parser BinomialDistribution
forall a b. (a -> b) -> a -> b
$Int -> Double -> Maybe BinomialDistribution
binomialE Int
n Double
p parseJSONValue
_=Parser BinomialDistribution
forall a. Parser a
forall (f :: * -> *) a. Alternative f => f a
emptyinstanceBinaryBinomialDistribution whereput :: BinomialDistribution -> Put
put (BD Int
x Double
y )=Int -> Put
forall t. Binary t => t -> Put
putInt
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forall a b. PutM a -> PutM b -> PutM b
forall (m :: * -> *) a b. Monad m => m a -> m b -> m b
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forall t. Binary t => t -> Put
putDouble
y get :: Get BinomialDistribution
get =doInt
n <-Get Int
forall t. Binary t => Get t
getDouble
p <-Get Double
forall t. Binary t => Get t
getGet BinomialDistribution
-> (BinomialDistribution -> Get BinomialDistribution)
-> Maybe BinomialDistribution
-> Get BinomialDistribution
forall b a. b -> (a -> b) -> Maybe a -> b
maybe([Char] -> Get BinomialDistribution
forall a. [Char] -> Get a
forall (m :: * -> *) a. MonadFail m => [Char] -> m a
fail([Char] -> Get BinomialDistribution)
-> [Char] -> Get BinomialDistribution
forall a b. (a -> b) -> a -> b
$Int -> Double -> [Char]
errMsg Int
n Double
p )BinomialDistribution -> Get BinomialDistribution
forall a. a -> Get a
forall (m :: * -> *) a. Monad m => a -> m a
return(Maybe BinomialDistribution -> Get BinomialDistribution)
-> Maybe BinomialDistribution -> Get BinomialDistribution
forall a b. (a -> b) -> a -> b
$Int -> Double -> Maybe BinomialDistribution
binomialE Int
n Double
p instanceD.Distribution BinomialDistribution wherecumulative :: BinomialDistribution -> Double -> Double
cumulative =BinomialDistribution -> Double -> Double
cumulative complCumulative :: BinomialDistribution -> Double -> Double
complCumulative =BinomialDistribution -> Double -> Double
complCumulative instanceD.DiscreteDistr BinomialDistribution whereprobability :: BinomialDistribution -> Int -> Double
probability =BinomialDistribution -> Int -> Double
probability logProbability :: BinomialDistribution -> Int -> Double
logProbability =BinomialDistribution -> Int -> Double
logProbability instanceD.Mean BinomialDistribution wheremean :: BinomialDistribution -> Double
mean =BinomialDistribution -> Double
mean instanceD.Variance BinomialDistribution wherevariance :: BinomialDistribution -> Double
variance =BinomialDistribution -> Double
variance instanceD.MaybeMean BinomialDistribution wheremaybeMean :: BinomialDistribution -> Maybe Double
maybeMean =Double -> Maybe Double
forall a. a -> Maybe a
Just(Double -> Maybe Double)
-> (BinomialDistribution -> Double)
-> BinomialDistribution
-> Maybe Double
forall b c a. (b -> c) -> (a -> b) -> a -> c
.BinomialDistribution -> Double
forall d. Mean d => d -> Double
D.mean instanceD.MaybeVariance BinomialDistribution wheremaybeStdDev :: BinomialDistribution -> Maybe Double
maybeStdDev =Double -> Maybe Double
forall a. a -> Maybe a
Just(Double -> Maybe Double)
-> (BinomialDistribution -> Double)
-> BinomialDistribution
-> Maybe Double
forall b c a. (b -> c) -> (a -> b) -> a -> c
.BinomialDistribution -> Double
forall d. Variance d => d -> Double
D.stdDev maybeVariance :: BinomialDistribution -> Maybe Double
maybeVariance =Double -> Maybe Double
forall a. a -> Maybe a
Just(Double -> Maybe Double)
-> (BinomialDistribution -> Double)
-> BinomialDistribution
-> Maybe Double
forall b c a. (b -> c) -> (a -> b) -> a -> c
.BinomialDistribution -> Double
forall d. Variance d => d -> Double
D.variance instanceD.Entropy BinomialDistribution whereentropy :: BinomialDistribution -> Double
entropy (BD Int
n Double
p )|Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
==Int
0=Double
0|Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<=Int
100=BinomialDistribution -> Double
directEntropy (Int -> Double -> BinomialDistribution
BD Int
n Double
p )|Bool
otherwise=Double -> Double
I.poissonEntropy (Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
n Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double
p )instanceD.MaybeEntropy BinomialDistribution wheremaybeEntropy :: BinomialDistribution -> Maybe Double
maybeEntropy =Double -> Maybe Double
forall a. a -> Maybe a
Just(Double -> Maybe Double)
-> (BinomialDistribution -> Double)
-> BinomialDistribution
-> Maybe Double
forall b c a. (b -> c) -> (a -> b) -> a -> c
.BinomialDistribution -> Double
forall d. Entropy d => d -> Double
D.entropy -- This could be slow for big nprobability ::BinomialDistribution ->Int->Doubleprobability :: BinomialDistribution -> Int -> Double
probability (BD Int
n Double
p )Int
k |Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
0Bool -> Bool -> Bool
||Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>Int
n =Double
0|Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
==Int
0=Double
1-- choose could overflow Double for n >= 1030 so we switch to-- log-domain to calculate probability---- We also want to avoid underflow when computing p^k &-- (1-p)^(n-k).|Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
1000,Double
pK Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
>=Double
m_tiny,Double
pNK Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
>=Double
m_tiny=Int -> Int -> Double
chooseInt
n Int
k Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double
pK Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double
pNK |Bool
otherwise=Double -> Double
forall a. Floating a => a -> a
exp(Double -> Double) -> Double -> Double
forall a b. (a -> b) -> a -> b
$Int -> Int -> Double
logChooseInt
n Int
k Double -> Double -> Double
forall a. Num a => a -> a -> a
+Double -> Double
forall a. Floating a => a -> a
logDouble
p Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double
k' Double -> Double -> Double
forall a. Num a => a -> a -> a
+Double -> Double
forall a. Floating a => a -> a
log1p(-Double
p )Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double
nk' wherepK :: Double
pK =Double
p Double -> Int -> Double
forall a b. (Num a, Integral b) => a -> b -> a
^Int
k pNK :: Double
pNK =(Double
1Double -> Double -> Double
forall a. Num a => a -> a -> a
-Double
p )Double -> Int -> Double
forall a b. (Num a, Integral b) => a -> b -> a
^(Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
k )k' :: Double
k' =Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
k nk' :: Double
nk' =Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegral(Int -> Double) -> Int -> Double
forall a b. (a -> b) -> a -> b
$Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
k logProbability ::BinomialDistribution ->Int->DoublelogProbability :: BinomialDistribution -> Int -> Double
logProbability (BD Int
n Double
p )Int
k |Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
0Bool -> Bool -> Bool
||Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>Int
n =(-Double
1)Double -> Double -> Double
forall a. Fractional a => a -> a -> a
/Double
0|Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
==Int
0=Double
0|Bool
otherwise=Int -> Int -> Double
logChooseInt
n Int
k Double -> Double -> Double
forall a. Num a => a -> a -> a
+Double -> Double
forall a. Floating a => a -> a
logDouble
p Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double
k' Double -> Double -> Double
forall a. Num a => a -> a -> a
+Double -> Double
forall a. Floating a => a -> a
log1p(-Double
p )Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double
nk' wherek' :: Double
k' =Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
k nk' :: Double
nk' =Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegral(Int -> Double) -> Int -> Double
forall a b. (a -> b) -> a -> b
$Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
k cumulative ::BinomialDistribution ->Double->Doublecumulative :: BinomialDistribution -> Double -> Double
cumulative (BD Int
n Double
p )Double
x |Double -> Bool
forall a. RealFloat a => a -> Bool
isNaNDouble
x =[Char] -> Double
forall a. HasCallStack => [Char] -> a
error[Char]
"Statistics.Distribution.Binomial.cumulative: NaN input"|Double -> Bool
forall a. RealFloat a => a -> Bool
isInfiniteDouble
x =ifDouble
x Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
>Double
0thenDouble
1elseDouble
0|Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
0=Double
0|Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>=Int
n =Double
1|Bool
otherwise=Double -> Double -> Double -> Double
incompleteBeta(Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegral(Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
k ))(Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegral(Int
k Int -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1))(Double
1Double -> Double -> Double
forall a. Num a => a -> a -> a
-Double
p )wherek :: Int
k =Double -> Int
forall b. Integral b => Double -> b
forall a b. (RealFrac a, Integral b) => a -> b
floorDouble
x complCumulative ::BinomialDistribution ->Double->DoublecomplCumulative :: BinomialDistribution -> Double -> Double
complCumulative (BD Int
n Double
p )Double
x |Double -> Bool
forall a. RealFloat a => a -> Bool
isNaNDouble
x =[Char] -> Double
forall a. HasCallStack => [Char] -> a
error[Char]
"Statistics.Distribution.Binomial.complCumulative: NaN input"|Double -> Bool
forall a. RealFloat a => a -> Bool
isInfiniteDouble
x =ifDouble
x Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
>Double
0thenDouble
0elseDouble
1|Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
0=Double
1|Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>=Int
n =Double
0|Bool
otherwise=Double -> Double -> Double -> Double
incompleteBeta(Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegral(Int
k Int -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1))(Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegral(Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
k ))Double
p wherek :: Int
k =Double -> Int
forall b. Integral b => Double -> b
forall a b. (RealFrac a, Integral b) => a -> b
floorDouble
x mean ::BinomialDistribution ->Doublemean :: BinomialDistribution -> Double
mean (BD Int
n Double
p )=Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
n Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double
p variance ::BinomialDistribution ->Doublevariance :: BinomialDistribution -> Double
variance (BD Int
n Double
p )=Int -> Double
forall a b. (Integral a, Num b) => a -> b
fromIntegralInt
n Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double
p Double -> Double -> Double
forall a. Num a => a -> a -> a
*(Double
1Double -> Double -> Double
forall a. Num a => a -> a -> a
-Double
p )directEntropy ::BinomialDistribution ->DoubledirectEntropy :: BinomialDistribution -> Double
directEntropy d :: BinomialDistribution
d @(BD Int
n Double
_)=Double -> Double
forall a. Num a => a -> a
negate(Double -> Double) -> ([Double] -> Double) -> [Double] -> Double
forall b c a. (b -> c) -> (a -> b) -> a -> c
.[Double] -> Double
forall a. Num a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
sum([Double] -> Double) -> [Double] -> Double
forall a b. (a -> b) -> a -> b
$(Double -> Bool) -> [Double] -> [Double]
forall a. (a -> Bool) -> [a] -> [a]
takeWhile(Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
<Double -> Double
forall a. Num a => a -> a
negateDouble
m_epsilon)([Double] -> [Double]) -> [Double] -> [Double]
forall a b. (a -> b) -> a -> b
$(Double -> Bool) -> [Double] -> [Double]
forall a. (a -> Bool) -> [a] -> [a]
dropWhile(Bool -> Bool
not(Bool -> Bool) -> (Double -> Bool) -> Double -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
.(Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
<Double -> Double
forall a. Num a => a -> a
negateDouble
m_epsilon))([Double] -> [Double]) -> [Double] -> [Double]
forall a b. (a -> b) -> a -> b
$[letx :: Double
x =BinomialDistribution -> Int -> Double
probability BinomialDistribution
d Int
k inDouble
x Double -> Double -> Double
forall a. Num a => a -> a -> a
*Double -> Double
forall a. Floating a => a -> a
logDouble
x |Int
k <-[Int
0..Int
n ]]-- | Construct binomial distribution. Number of trials must be-- non-negative and probability must be in [0,1] rangebinomial ::Int-- ^ Number of trials.->Double-- ^ Probability.->BinomialDistribution binomial :: Int -> Double -> BinomialDistribution
binomial Int
n Double
p =BinomialDistribution
-> (BinomialDistribution -> BinomialDistribution)
-> Maybe BinomialDistribution
-> BinomialDistribution
forall b a. b -> (a -> b) -> Maybe a -> b
maybe([Char] -> BinomialDistribution
forall a. HasCallStack => [Char] -> a
error([Char] -> BinomialDistribution) -> [Char] -> BinomialDistribution
forall a b. (a -> b) -> a -> b
$Int -> Double -> [Char]
errMsg Int
n Double
p )BinomialDistribution -> BinomialDistribution
forall a. a -> a
id(Maybe BinomialDistribution -> BinomialDistribution)
-> Maybe BinomialDistribution -> BinomialDistribution
forall a b. (a -> b) -> a -> b
$Int -> Double -> Maybe BinomialDistribution
binomialE Int
n Double
p -- | Construct binomial distribution. Number of trials must be-- non-negative and probability must be in [0,1] rangebinomialE ::Int-- ^ Number of trials.->Double-- ^ Probability.->MaybeBinomialDistribution binomialE :: Int -> Double -> Maybe BinomialDistribution
binomialE Int
n Double
p |Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<Int
0=Maybe BinomialDistribution
forall a. Maybe a
Nothing|Double
p Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
>=Double
0Bool -> Bool -> Bool
&&Double
p Double -> Double -> Bool
forall a. Ord a => a -> a -> Bool
<=Double
1=BinomialDistribution -> Maybe BinomialDistribution
forall a. a -> Maybe a
Just(Int -> Double -> BinomialDistribution
BD Int
n Double
p )|Bool
otherwise=Maybe BinomialDistribution
forall a. Maybe a
NothingerrMsg ::Int->Double->StringerrMsg :: Int -> Double -> [Char]
errMsg Int
n Double
p =[Char]
"Statistics.Distribution.Binomial.binomial: n="[Char] -> ShowS
forall a. [a] -> [a] -> [a]
++Int -> [Char]
forall a. Show a => a -> [Char]
showInt
n [Char] -> ShowS
forall a. [a] -> [a] -> [a]
++[Char]
" p="[Char] -> ShowS
forall a. [a] -> [a] -> [a]
++Double -> [Char]
forall a. Show a => a -> [Char]
showDouble
p [Char] -> ShowS
forall a. [a] -> [a] -> [a]
++[Char]
"but n>=0 and p in [0,1]"