Factory.Math.Probability

AUTHOR

Dr. Alistair Ward

DESCRIPTION

Miscellaneous functions for probability-distributions.

Types

Data-types

data ContinuousDistribution f

Describes a continuous probability-distribution; http://en.wikipedia.org/wiki/List_of_probability_distributions#Continuous_distributions.

Constructors

UniformDistribution (Interval f)

Defines a Uniform-distribution within a closed interval; http://en.wikipedia.org/wiki/Uniform_distribution.

NormalDistribution f f

Defines a Normal-distribution with a particular mean and variance; http://en.wikipedia.org/wiki/Normal_distribution.

Instances

Eq f => Eq (ContinuousDistribution f)

Read f => Read (ContinuousDistribution f)

Show f => Show (ContinuousDistribution f)

(Num a, Ord a, Show a) => SelfValidator (ContinuousDistribution a)

data DiscreteDistribution f

Describes a discrete probability-distribution; http://en.wikipedia.org/wiki/List_of_probability_distributions#Discrete_distributions.

Constructors

PoissonDistribution f

Instances

Eq f => Eq (DiscreteDistribution f)

Read f => Read (DiscreteDistribution f)

Show f => Show (DiscreteDistribution f)

(Num f, Ord f, Show f) => SelfValidator (DiscreteDistribution f)

Functions

boxMullerTransform

Arguments

:: (Floating f, Ord f, Show f)

=> (f, f)

Independent, uniformly distributed random numbers, which must be within the semi-closed unit interval, (0, 1].

-> (f, f)

Independent, normally distributed random numbers, with standardized mean=0 and variance=1.

• Converts a pair of independent uniformly distributed random numbers, within the semi-closed unit interval (0 .. 1], to a pair of independent normally distributed random numbers, of standardized mean=0, and variance=1.
• http://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform.

generateStandardizedNormalDistribution :: (RealFloat f, Show f, Random f, RandomGen randomGen) => randomGen -> [f]

• Uses the supplied random-number generator, to generate a conceptually infinite list, of normally distributed random numbers, with standardized mean=0, and variance=1.
• http://en.wikipedia.org/wiki/Normal_distribution, http://mathworld.wolfram.com/NormalDistribution.html.

generateContinuousPopulation

Arguments

:: (RealFloat f, Show f, Random f, RandomGen randomGen)

=> Int

number of items.

-> ContinuousDistribution f

-> randomGen

A generator of uniformly distributed random numbers.

-> [f]

• Generates a random sample-population, with the specified continuous probability-distribution.
• When a Normal distribution is requested, the generated population will only tend towards the requested mean and variance of, as the sample-size tends towards infinity. Whilst one could arrange for these criteria to be precisely met for any sample-size, the sample would lose a degree of randomness as a result.

generatePoissonDistribution

Arguments

:: (Integral events, RealFloat lambda, Show lambda, Random lambda, RandomGen randomGen)

=> lambda

Defines the required approximate value of both mean and variance.

-> randomGen

-> [events]

• Uses the supplied random-number generator, to generate a conceptually infinite list, of random integers conforming to the Poisson distribution (mean=lambda, variance=lambda).
• http://en.wikipedia.org/wiki/Poisson_distribution.
• CAVEAT: uses an algorithm by Knuth, which having a linear time-complexity in lambda, can be intolerably slow; also, the term exp $ negate lambda, underflows for large lambda; so for large lambda, this implementation returns the appropriate NormalDistribution, which is similar for large lambda.

generateDiscretePopulation

Arguments

:: (Ord f, RealFloat f, Show f, Random f, RandomGen randomGen, Integral events)

=> Int

number of items.

-> DiscreteDistribution f

-> randomGen

A generator of uniformly distributed random numbers.

-> [events]

Generates a random sample-population, with the specified discrete probability-distribution.