module Alea::CDF

Overview

Alea::CDF is the interface used to calculate the Cumulative Distribution Functions. Given X ~ D and a fixed quantile x, CDFs are defined as the functions that associate x to the probability that the real-valued random X from the distribution D will take a value less or equal to x.

Arguments passed to Alea::CDF methods to shape the distributions are analogous to those used for sampling:

Alea::CDF.normal(0.0)                       # => 0.5
Alea::CDF.normal(2.0, loc: 1.0, sigma: 0.5) # => 0.9772498680518208
Alea::CDF.chisq(5.279, df: 5.0)             # => 0.6172121213841358

NOTE for real-valued quantiles is used x, k for discrete-valued instead.

Defined in:

alea/prob/cdf.cr
alea/prob/pchisq.cr
alea/prob/pexp.cr
alea/prob/pgamma.cr
alea/prob/plaplace.cr
alea/prob/plognor.cr
alea/prob/pnorm.cr
alea/prob/ppoiss.cr
alea/prob/punif.cr

Class Method Summary

.chisq(x, df) : Float64
Calculate the cumulative distribution function evaluated at x.
.chisq32(x, df) : Float32
Calculate the cumulative distribution function evaluated at x.
.exp(x, scale = 1.0) : Float64
Calculate the cumulative distribution function evaluated at x.
.exp32(x, scale = 1.0_f32) : Float32
Calculate the cumulative distribution function evaluated at x.
.gamma(x, shape, scale = 1.0) : Float64
Calculate the cumulative distribution function evaluated at x.
.gamma32(x, shape, scale = 1.0_f32) : Float32
Calculate the cumulative distribution function evaluated at x.
.laplace(x, loc = 0.0, scale = 1.0) : Float64
Calculate the cumulative distribution function evaluated at x.
.laplace32(x, loc = 0.0_f32, scale = 1.0_f32) : Float32
Calculate the cumulative distribution function evaluated at x.
.lognormal(x, loc = 0.0, sigma = 1.0) : Float64
Calculate the cumulative distribution function evaluated at x.
.lognormal32(x, loc = 0.0_f32, sigma = 1.0_f32) : Float32
Calculate the cumulative distribution function evaluated at x.
.normal(x, loc = 0.0, sigma = 1.0) : Float64
Calculate the cumulative distribution function evaluated at x.
.normal32(x, loc = 0.0_f32, sigma = 1.0_f32) : Float32
Calculate the cumulative distribution function evaluated at x.
.poisson(k, lam = 1.0) : Float64
Calculate the cumulative distribution function evaluated at k.
.poisson32(k, lam = 1.0_f32) : Float32
Calculate the cumulative distribution function evaluated at k.
.uniform(x, min, max) : Float64
Calculate the cumulative distribution function evaluated at x.
.uniform32(x, min, max) : Float32
Calculate the cumulative distribution function evaluated at x.

Class Method Detail

def self.chisq(x, df) : Float64 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
df: degrees of freedom of the distribution; usually mentioned as k.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if df is negative or zero.

[View source]

def self.chisq32(x, df) : Float32 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
df: degrees of freedom of the distribution; usually mentioned as k.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if df is negative or zero.

[View source]

def self.exp(x, scale = 1.0) : Float64 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
scale: scale parameter of the distribution; usually mentioned as λ^-1.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if scale is negative or zero.

[View source]

def self.exp32(x, scale = 1.0_f32) : Float32 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
scale: scale parameter of the distribution; usually mentioned as λ^-1.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if scale is negative or zero.

[View source]

def self.gamma(x, shape, scale = 1.0) : Float64 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
shape: shape parameter of the distribution; usually mentioned as k.
scale: scale parameter of the distribution; usually mentioned as θ.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if any of shape or scale is negative or zero.

[View source]

def self.gamma32(x, shape, scale = 1.0_f32) : Float32 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
shape: shape parameter of the distribution; usually mentioned as k.
scale: scale parameter of the distribution; usually mentioned as θ.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if any of shape or scale is negative or zero.

[View source]

def self.laplace(x, loc = 0.0, scale = 1.0) : Float64 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
loc: centrality parameter, or mean of the distribution; usually mentioned as μ.
scale: scale parameter of the distribution; usually mentioned as b.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if scale is negative or zero.

[View source]

def self.laplace32(x, loc = 0.0_f32, scale = 1.0_f32) : Float32 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
loc: centrality parameter, or mean of the distribution; usually mentioned as μ.
scale: scale parameter of the distribution; usually mentioned as b.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if scale is negative or zero.

[View source]

def self.lognormal(x, loc = 0.0, sigma = 1.0) : Float64 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
loc: centrality parameter, or mean of the underlying normal distribution; usually mentioned as μ.
sigma: scale parameter, or standard deviation of the underlying normal distribution; usually mentioned as σ.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if sigma is negative or zero.

[View source]

def self.lognormal32(x, loc = 0.0_f32, sigma = 1.0_f32) : Float32 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
loc: centrality parameter, or mean of the underlying normal distribution; usually mentioned as μ.
sigma: scale parameter, or standard deviation of the underlying normal distribution; usually mentioned as σ.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if sigma is negative or zero.

[View source]

def self.normal(x, loc = 0.0, sigma = 1.0) : Float64 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
loc: centrality parameter, or mean of the distribution; usually mentioned as μ.
sigma: scale parameter, or standard deviation of the distribution; usually mentioned as σ.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if sigma is negative or zero.

[View source]

def self.normal32(x, loc = 0.0_f32, sigma = 1.0_f32) : Float32 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
loc: centrality parameter, or mean of the distribution; usually mentioned as μ.
sigma: scale parameter, or standard deviation of the distribution; usually mentioned as σ.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if sigma is negative or zero.

[View source]

def self.poisson(k, lam = 1.0) : Float64 #

Calculate the cumulative distribution function evaluated at k.

@parameters:

k: discrete-valued quantile of which estimate the cdf.
lam: separation parameter of the distribution; usually mentioned as λ.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if lam is negative or zero.

[View source]

def self.poisson32(k, lam = 1.0_f32) : Float32 #

Calculate the cumulative distribution function evaluated at k.

@parameters:

k: discrete-valued quantile of which estimate the cdf.
lam: separation parameter of the distribution; usually mentioned as λ.

@exceptions:

Alea::NaNError if any of the arguments is NaN.
Alea::InfinityError if any of the arguments is Infinity.
Alea::UndefinedError if lam is negative or zero.

[View source]

def self.uniform(x, min, max) : Float64 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
min: left bound parameter of range of the distribution; usually mentioned as a.
max: right bound parameter of range of the distribution; usually mentioned as b.

@notes:

range is [min, max).

@exceptions:

Alea::NaNError if any of the arguments bound is NaN.
Alea::InfinityError if any of the arguments bound is Infinity.
Alea::UndefinedError if max is less than min.

[View source]

def self.uniform32(x, min, max) : Float32 #

Calculate the cumulative distribution function evaluated at x.

@parameters:

x: real-valued quantile of which estimate the cdf.
min: left bound parameter of range of the distribution; usually mentioned as a.
max: right bound parameter of range of the distribution; usually mentioned as b.

@notes:

range is [min, max).

@exceptions:

Alea::NaNError if any of the arguments bound is NaN.
Alea::InfinityError if any of the arguments bound is Infinity.
Alea::UndefinedError if max is less than min.

[View source]

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alea

module Alea::CDF

Overview

Defined in:

Class Method Summary

Class Method Detail