source.univariate package

This package a collection of continuous univariate distributions.

source.univariate.BoundedInterval module

class source.univariate.BoundedInterval.Arcsine

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning Arcsine Distirbution 1.

\[\text{Arcsine}(x)={\frac{1}{\pi \sqrt{x(1-x)}}}\]

Parameters

x (float) -- random variable between 0 and 1

Reference:

1

Wikipedia contributors. (2020, October 30). Arcsine distribution. https://en.wikipedia.org/w/index.php?title=Arcsine_distribution&oldid=986131091

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point

Raises

• ValueError -- when there exist a value less than 0 or greater than 1

• TypeError -- when parameter is not of type float | List[float] | numpy.ndarray

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns

entropy of Arcsine distribution

kurtosis() → float

Returns

kurtosis of Arcsine distribution

mean() → float

Returns

mean of Arcsine distribution.

median() → float

Returns

median of Arcsine distribution

mode() → Tuple[float, float]

Returns

mode of Arcsine distribution

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variables

Raises

• ValueError -- when there exist a value less than 0 or greater than 1

• TypeError -- when parameter is not of type float | List[float] | numpy.ndarray

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns

skewness of Arcsine distribution

std() → float

Returns

standard deviation of Arcsine distribution

summary() → Dict[str, Union[float, Tuple[float, float]]]

Returns

Dictionary of Arcsine distirbution moments. This includes standard deviation.

var() → float

Returns

variance of Arcsine distribution

class source.univariate.BoundedInterval.Bates(a: float, b: float, n: int, randvar: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning Bates Distirbution. Also referred to as the regular mean distribution.

Note that the Bates distribution is a probability distribution of the mean of a number of statistically indipendent uniformly distirbuted random variables on the unit interval. This is often confused with the Irwin-Hall distribution which is the distribution of the sum (not the mean) of n independent random variables. The two distributions are simply versions of each other as they only differ in scale 2.

Parameters

• a (float) -- lower bound parameter

• b (float) -- upper bound parameter where b > a

• n (int) -- where n >= 1

• randvar (float) -- random variable where a <= x <= b

Reference:

2

Wikipedia contributors. (2021, January 8). Bates distribution. https://en.wikipedia.org/w/index.php?title=Bates_distribution&oldid=999042206

cdf(x: Optional[Union[List[float], numpy.ndarray]] = None) → Union[float, numpy.ndarray]

Parameters

x (List[float], numpy.ndarray) -- random variable or list of random variables

Returns

either cumulative distribution evaluation for some point or plot of Bates distribution.

kurtosis() → float

Returns: Kurtosis of the Bates distribution.

mean() → float

Returns: Mean of the Bates distribution.

skewness() → float

Returns: Skewness of the Bates distribution.

std() → float

Returns: Standard devtiation of the Bates distribution

summary() → Dict[str, Union[float, Tuple[float]]]

Summary statistic regarding the Bates distribution which contains the following parts of the distribution: (mean, median, mode, var, std, skewness, kurtosis).

Returns

[description]

Return type

Dict[str, Union[float, Tuple[float]]]

var() → float

Returns: Variance of the Bates distribution.

class source.univariate.BoundedInterval.Bernoulli(shape: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning Continuous Bernoulli Distirbution. The continuous Bernoulli distribution arises in deep learning and computer vision, specifically in the context of variational autoencoders, for modeling the pixel intensities of natural images 3 4 5 6.

\[C(\lambda)\lambda^{x}(1-\lambda)^{1-x}\]

where

\[C(\lambda)= \begin{cases}2&{\text{if }\lambda =\frac {1}{2}} \ \frac{2\tanh^{-1}(1-2\lambda )}{1-2\lambda }&{\text{ otherwise}}\end{cases}\]

Parameters

• shape (float) -- parameter

• x (float) -- random variable where x is between 0 and 1

Reference:

3

Wikipedia contributors. (2020, November 2). Continuous Bernoulli distribution. https://en.wikipedia.org/w/index.php?title=Continuous_Bernoulli_distribution&oldid=986761458

4

Kingma, D. P., & Welling, M. (2013). Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114.

5

Kingma, D. P., & Welling, M. (2014, April). Stochastic gradient VB and the variational auto-encoder.In Second International Conference on Learning Representations, ICLR (Vol. 19).

6

Ganem, G & Cunningham, J.P. (2019). The continouous Bernoulli: fixing a pervasive error in variational autoencoders. https://arxiv.org/pdf/1907.06845.pdf

cdf(x: Union[List[float], numpy.ndarray]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray]) -- data point(s) of interest

Raises

ValueError -- when there exist a value <= 0 or >= 1

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

mean() → float

Returns: Mean of the Continuous Bernoulli distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value less than 0 or greater than 1

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

std() → float

Returns: Standard deviation of the Continuous Bernoulli distribution

summary() → Dict[str, Union[float, Tuple[float]]]

Returns

Dictionary of Continuous Bernoulli distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Continuous Bernoulli distribution.

class source.univariate.BoundedInterval.Beta(alpha: float, beta: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning Beta Distirbution 7.

\[\text{Beta}(x; \alpha, \beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\text{B}(\alpha, \beta)}\]

Parameters

• alpha (float) -- shape parameter where alpha > 0

• beta (float) -- shape parameter where beta > 0

• x (float) -- random variable where x is between 0 and 1

Reference:

7

Wikipedia contributors. (2021, January 8). Beta distribution. https://en.wikipedia.org/w/index.php?title=Beta_distribution&oldid=999043368

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Beta distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Beta distribution.

mean() → str

Returns: Mean of the Beta distribution.

median() → float

Returns: Median of the Beta distribution.

mode() → str

Returns: Mode of the Beta distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value x <= 0 or x <= 1

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Beta distribution.

std() → str

Returns: Variance of the Beta distribution.

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Beta distirbution moments. This includes standard deviation.

var() → str

Returns: Variance of the Beta distribution.

class source.univariate.BoundedInterval.BetaRectangular(alpha: float, beta: float, theta: float, min: float, max: float, randvar: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning Beta-rectangular Distirbution. Thus it is a bounded distribution that allows for outliers to have a greater chance of occurring than does the beta distribution 8 .

\[\begin{split}\text{BetaRectangulat}(x,\alpha ,\beta ,\theta )={\begin{cases}{\frac{\theta \Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}}{\frac{(x-a)^{{\alpha -1}}(b-x)^{{\beta -1}}}{(b-a)^{{\alpha +\beta +1}}}}+{\frac{1-\theta }{b-a}}&{\mathrm{for}}\ a\leq x\leq b,\\[8pt]0&{\mathrm{for}}\ x<a\{\mathrm{or}}\ x>b\end{cases}}\end{split}\]

Parameters

• alpha (float) -- shape parameter

• beta (float) -- shape parameter

• theta (float) -- mixture parameter where 0 < theta < 1

• min (float) -- lower bound

• max (float) -- upper bound

• x (float) -- random variable where alpha <= x<= beta

Reference:

8

Wikipedia contributors. (2020, December 7). Beta rectangular distribution. https://en.wikipedia.org/w/index.php?title=Beta_rectangular_distribution&oldid=992814814

cdf()

mean() → float

Returns: Mean of the Beta-rectangular distribution.

pdf()

std() → float

Returns: Standard deviation of the Beta-rectangular distribution.

summary() → Dict[str, Union[float, Tuple[float]]]

Summary statistic regarding the Beta-rectangular distribution which contains the following parts of the distribution: (mean, median, mode, var, std, skewness, kurtosis).

var() → float

Returns: Variance of the Beta-rectangular distribution.

class source.univariate.BoundedInterval.BoundedInterval

Bases: univariate._base.Base

Description:

Base class for probability tags.

class source.univariate.BoundedInterval.Kumaraswamy(a: float, b: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning the Kumaraswamy distribution 9 .

\[\text{Kumaraswamy}(x;a,b) = abx^{a-1}(1-x^a)^{b-1}\]

Parameters

• a (float) -- a parameter \(a > 0\)

• b (float) -- b parameter \(b > 0\)

• x (float) -- random variables \(x \in (0,1)\)

Reference:

9

Wikipedia Contributors (2021). Kumaraswamy distribution. https://en.wikipedia.org/wiki/Kumaraswamy_distribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Raises

ValueError -- when there exist a value of x <= 0 or x >= 1

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Default implementation of entropy. Returns NotImplemented.

kurtosis() → float

Default implementation of kurtosis. Returns NotImplemented.

mean() → float

Default implementation of the mean. Returns NotImplemented.

median() → float

Default implementation of the median. Returns NotImplemented.

mode() → Union[float, str]

Default implementation of the mode. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x <= 0 or x>= 1

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Default implementation of skewness. Returns NotImplemented.

std() → float

Default implementation of the standard deviation. Returns NotImplemented.

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Kumaraswamy distirbution moments. This includes standard deviation.

var() → float

Default implementation of the variance. Returns NotImplemented.

class source.univariate.BoundedInterval.LogitNormal(sq_scale: float, location: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning Logit Normal Distirbution 10.

\[\text{LogitNormal}(x;\mu,\sigma) = \frac{1}{\sigma \sqrt{2\pi} \cdot x(1-x)} \exp{\Big(-\frac{(logit(x)-\mu)^2}{2\sigma^2} \Big)}\]

Parameters

• sq_scale (float) -- squared scale parameter

• location (float) -- location parameter

• x (float) -- random variable where x is between 0 and 1

Reference:

10

Wikipedia contributors. (2020, December 9). Logit-normal distribution. https://en.wikipedia.org/w/index.php?title=Logit-normal_distribution&oldid=993237113

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → str

Returns: differential entropy of Logit Normal distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

mean() → str

Returns: Mean of the Logit Normal distribution.

mode() → str

Returns: Mode of the Logit Normal distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value below 0 and greater than 1

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

std() → str

Returns: Standard deviation of the Logit Normal distribution.

summary() → Dict[str, str]

Returns

Dictionary of Logit Normal distirbution moments. This includes standard deviation.

var() → str

Returns: Variance of the Logit Normal distribution.

class source.univariate.BoundedInterval.RaisedCosine(mu: float, s: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning the Raised Cosine distribution 11.

\[\text{RaisedCosine}(x;\mu, s) = \frac{1}{2s} \Big[ 1 + \cos \Big( \frac{x-\mu}{s} \pi \Big) \Big]\]

Parameters

• mu (float) -- mu parameter \(\mu \in \mathbb{R}\)

• s (float) -- s parameter \(s > 0\)

• x (float) -- random variables

References

11

Wikipedia Contributors (2020). Raised cosine distribution. https://en.wikipedia.org/wiki/Raised_cosine_distribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

kurtosis() → float

Default implementation of kurtosis. Returns NotImplemented.

mean() → float

Default implementation of the mean. Returns NotImplemented.

median() → float

Default implementation of the median. Returns NotImplemented.

mode() → float

Default implementation of the mode. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x < mu - s or x > mu + s

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Default implementation of skewness. Returns NotImplemented.

std() → float

Default implementation of the standard deviation. Returns NotImplemented.

summary() → Dict[str, float]

Returns

Dictionary of Raised Cosine distirbution moments. This includes standard deviation.

var() → float

Default implementation of the variance. Returns NotImplemented.

class source.univariate.BoundedInterval.Reciprocal(a: float, b: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning Reciprocal distribution 12.

\[\text{Reciprocal}(x;a,b) = \frac{1}{x \ln \Big( \frac{b}{a}\Big)}\]

Parameters

• a (float) -- a parameter \(a > 0\)

• b (float) -- b prameter \(b > 0\)

• x (float) -- random variable

Reference:

12

Wikipedia Contributors (2021). Reciprocal distribution. https://en.wikipedia.org/wiki/Reciprocal_distribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

mean() → float

Default implementation of the mean. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

std() → float

Default implementation of the standard deviation. Returns NotImplemented.

summary() → Dict[str, float]

Returns

Dictionary of Reciprocal distirbution moments. This includes standard deviation.

variance() → float

class source.univariate.BoundedInterval.Trapezoidal(a: float, b: float, c: float, d: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning Trapezoidal Distirbution 13.

Parameters

• a (float) -- lower bound parameter where a < d

• b (float) -- level start parameter where a <= b < c

• c (float) -- level end parameter where b < c <= d

• d (float) -- upper bound parameter where c <= d

• randvar (float) -- random variable where a <= x <= d

Reference:

13

Wikipedia contributors. (2020, April 11). Trapezoidal distribution. https://en.wikipedia.org/w/index.php?title=Trapezoidal_distribution&oldid=950241388

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

mean() → float

Returns: Mean of the Trapezoidal distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

summary() → Dict[str, Union[float, Tuple[float]]]

Returns

Dictionary of Trapezoidal distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Trapezoidal distribution. Currently Unsupported.

class source.univariate.BoundedInterval.Triangular(a: float, b: float, c: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning Triangular Distirbution 14.

Parameters

• a (float) -- lower limit parameter

• b (float) -- upper limit parameter where a < b

• c (float) -- mode parameter where a <= c <= b

• randvar (float) -- random variable where a <= x <= b

Reference:

14

Wikipedia contributors. (2020, December 19). Triangular distribution. https://en.wikipedia.org/w/index.php?title=Triangular_distribution&oldid=995101682

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation fo cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Triangular distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Triangular distribution.

mean() → float

Returns: Mean of the Triangular distribution.

median() → float

Returns: Median of the Triangular distribution.

mode() → float

Returns: Mode of the Triangular distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of a > x or x > b

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Triangular distribution.

std() → float

Returns: Standard deviation of the Triangular distribution.

summary() → Dict[str, float]

Returns

Dictionary of Triangular distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Triangular distribution.

class source.univariate.BoundedInterval.UQuadratic(a: float, b: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning U-Quadratic Distribution 15.

\[\text{UQuadratic}(x;a,b) = a \Big(x- \frac{a+b}{2}\Big)^2\]

Parameters

• a (float) -- parameter a

• b (float) -- parameter b

• x (float) -- random variables \(x \in [a,b]\)

Reference:

15

Wikipedia Contributors (2021). U-quadratic distribution. https://en.wikipedia.org/wiki/U-quadratic_distribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Raises

ValueError -- when there exist a value of x < a or x > b

Returns

cdf evaluation at x

Return type

Union[float, numpy.ndarray]

kurtosis() → float

Default implementation of kurtosis. Returns NotImplemented.

mean() → float

Default implementation of the mean. Returns NotImplemented.

median() → float

Default implementation of the median. Returns NotImplemented.

mode() → Tuple[float, float]

Default implementation of the mode. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x less than a or greater than b

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Default implementation of skewness. Returns NotImplemented.

std() → float

Default implementation of the standard deviation. Returns NotImplemented.

var() → float

Default implementation of the variance. Returns NotImplemented.

class source.univariate.BoundedInterval.Uniform(a: float, b: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning the Continuous Uniform Distribution 16.

\[\text{Uniform}(x;a,b) = \frac{1}{b-a}\]

Parameters

• a (float) -- lower limit of the distribution

• b (float) -- upper limit of the distribution where b > a

Referene:

16

Weisstein, Eric W. "Uniform Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UniformDistribution.html

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: entropy of uniform Distirbution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Uniform distribution.

mean() → float

Returns: Mean of the Uniform distribution.

median() → float

Returns: Median of the Uniform distribution.

mode() → Tuple[float, float]

Returns: Mode of the Uniform distribution.

Note that the mode is any value in (a,b)

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evauation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Uniform distribution.

std() → float

Returns: Standard deviation of the Uniform distribution.

summary() → Dict[str, Union[float, Tuple[float, float]]]

Returns

Dictionary of Uniform distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Uniform distribution.

class source.univariate.BoundedInterval.WignerSemiCircle(radius: float)

Bases: source.univariate.BoundedInterval.BoundedInterval

This class contains methods concerning the Wigner Semi Circle 17.

\[\text{WignerSemiCircle}(x;r) = \frac{2}{ \pi r^2} \sqrt{r^2 - x^2}\]

Parameters

• radius (float) -- raduis parameter \(r > 0\)

• x (float) -- random variable

Reference:

17

Wikipedia Contributors (2021). Wigner semicircle distribution. https://en.wikipedia.org/wiki/Wigner_semicircle_distribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Raises

ValueError -- when there exist a value of x < -rad or x > rad

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Default implementation of entropy. Returns NotImplemented.

kurtorsis() → float

mean() → float

Default implementation of the mean. Returns NotImplemented.

median() → float

Default implementation of the median. Returns NotImplemented.

mode() → float

Default implementation of the mode. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variables

Raises

ValueError -- when there exist a value of x < -rad or x > rad

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Default implementation of skewness. Returns NotImplemented.

std() → float

Default implementation of the standard deviation. Returns NotImplemented.

summary() → Dict[str, float]

Returns

Dictionary of WignerSemiCircle distirbution moments. This includes standard deviation.

var() → float

Default implementation of the variance. Returns NotImplemented.

source.univariate.Infinite module

class source.univariate.Infinite.AssymetricLaplace(loc: float, scale: float, asym: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Assymetric Laplace Ditribution 18.

\[{\displaystyle \text{AssymetricLaplace}(x;m,\lambda ,\kappa )={\]

rac {lambda }{kappa +1/kappa }}{begin{cases}exp left((lambda /kappa )(x-m)right)&{text{if }}x<m[4pt]exp(-lambda kappa (x-m))&{text{if }}xgeq mend{cases}}}

Args:

loc (float): location parameter \(m\) scale (float): scale parameter \(\lambda > 0\) asym (float): assymetry parameter \(\kappa > 0\) x (float): random variable

Reference:

18

Wikipedia Contributors (2020). Assymetric Laplace Distribution. https://en.wikipedia.org/wiki/Asymmetric_Laplace_distribution

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: entropy of the Assymetric Laplace distribution.

kurtosis() → float

Returns: Kurtosis of the Assymetric Laplace distribution.

mean() → float

Returns: Mean of the Assymetric Laplace distribution.

median() → Optional[float]

Returns: Median of the Assymetric Laplace distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Assymetric Laplace distribution.

std() → float

Returns: Standard Deviation of the Assymetric Laplace distribution.

summary() → Dict[str, Optional[float]]

Returns

Dictionary of Assymetric Laplace distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Assymetric Laplace distribution.

class source.univariate.Infinite.Cauchy(loc: float, scale: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning the Cauchy Distribution 19 20.

\[\text{Cauchy}(x;loc, scale) = \frac{1}{\pi \cdot scale \big[ 1 + \big( \frac{x-loc}{scale} \big)^2 \big]}\]

Parameters

• loc (float) -- pertains to the loc parameter or median

• scale (float) -- pertains to the scale parameter where scale > 0

• x (float) -- random variable

References

19

Wikipedia contributors. (2020, November 29). Cauchy distribution. https://en.wikipedia.org/w/index.php?title=Cauchy_distribution&oldid=991234690

20

Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchyDistribution.html

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Cauchy distribution

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Cauchy distribution

mean() → str

Returns: Mean of the Cauchy distribution. Mean is Undefined.

median() → float

Returns: Median of the Cauchy distribution.

mode() → float

Returns: Mode of the Cauchy distribution

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → str

Returns: Skewness of the Cauchy distribution.

std() → str

Returns: Standard Deviation of the Cauchy Distribution.

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Cauchy distirbution moments. This includes standard deviation.

var() → str

Returns: Variance of the Cauchy distribution.

class source.univariate.Infinite.FisherZ(df1: float, df2: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning Fisher's z-Distribution 21 22 23.

\[\text{Fisher}(x;d_1, d_2) = \frac{2d_2^{d_1/2} d_2^{d_2/2}}{\text{B}\Big(\frac{d_1}{2}, \frac{d_2}{2}\Big)} \frac{e^{d_1 x}}{(d_1 e^{2x} + d_2)^{(d_1+d_2)/2}}\]

Parameters

• df1 (float) -- degrees of freedom (\(d_1 > 0\)).

• df2 (float) -- degrees of freedom (\(d_2 > 0\)).

• x (float) -- random variable.

Note: Fisher's z-distribution is the statistical distribution of half the log of an F-distribution variate: z = 1/2*log(F)

Reference:

21

Wikipedia contributors. (2020, December 15). Fisher's z-distribution. https://en.wikipedia.org/w/index.php?title=Fisher%27s_z-distribution&oldid=994427156.

22

Wolfram Research (2010), FisherZDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FisherZDistribution.html (updated 2016).

23

Wolfram Alpha (2021). Fisher Distribution. https://www.wolframalpha.com/input/?i=Fisher+distribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], np.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, np.ndarray]

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

summary() → Dict[str, float]

Returns

Dictionary of Fisher z distirbution moments. This includes standard deviation.

class source.univariate.Infinite.GH(lmbda: float, alpha: float, asym: float, scale: float, loc: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Generalized Hyperbolic Distribution V1 24.

Parameters

• alpha (float) -- alpha parameter \(\alpha\)

• lmbda (float) -- lambda parameter \(\lambda\)

• asym (float) -- asymmetry parameter \(\beta\)

• scale (float) -- scale parameter \(\delta\)

• loc (float) -- location parameter \(\mu\)

• x (float) -- random variable

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

mean() → float

Default implementation of the mean. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

std() → float

Default implementation of the standard deviation. Returns NotImplemented.

summary() → Dict[str, float]

Returns

Dictionary of GH distirbution moments. This includes standard deviation.

var() → float

Default implementation of the variance. Returns NotImplemented.

class source.univariate.Infinite.GNV1(loc: float, scale: float, shape: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Generalized Normal Distribution V1 25.

\[\text{GNV1}(x; \mu, \alpha, \beta) = \frac{\beta}{2 \alpha \Gamma(1/\beta)} e^{(- |x-\mu|/ \alpha)^\beta}\]

Parameters

• loc (float) -- location parameter \(\mu\)

• scale (float) -- scale parameter \(\alpha\)

• shape (float) -- shape parameter \(\beta\)

• x (float) -- random variable

Reference:

24

Wikipedia Contributors (2021). Generalized normal distribution. https://en.wikipedia.org/wiki/Generalized_normal_distribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

entropy() → float

Returns: Entropy of the GNV1 distribution.

kurtosis() → float

Returns: Kurtosis of the GNV1 distribution.

mean() → float

Returns: Mean of the GNV1 distribution.

median() → float

Returns: Median of the GNV1 distribution.

mode() → float

Returns: Mode of the GNV1 distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the GNV1 distribution.

std() → float

Returns: Standard Deviation of the GNV1 distribution.

summary() → Dict[str, Optional[float]]

Returns

Dictionary of GNV1 distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the GNV1 distribution.

class source.univariate.Infinite.GNV2(loc: float, scale: float, shape: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Generalized Normal Distribution V2 26.

Parameters

• loc (float) -- location parameter \(\xi\)

• scale (float) -- scale parameter \(\alpha\)

• shape (float) -- shape parameter \(\beta\)

• x (float) -- random variable

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

entropy() → float

Default implementation of entropy. Returns NotImplemented.

kurtosis() → float

Default implementation of kurtosis. Returns NotImplemented.

mean() → float

Default implementation of the mean. Returns NotImplemented.

median() → float

Default implementation of the median. Returns NotImplemented.

mode() → float

Default implementation of the mode. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

skewness() → float

Default implementation of skewness. Returns NotImplemented.

std() → float

Default implementation of the standard deviation. Returns NotImplemented.

summary() → Dict[str, float]

Returns

Dictionary of GNV2 distirbution moments. This includes standard deviation.

var() → float

Default implementation of the variance. Returns NotImplemented.

class source.univariate.Infinite.Gaussian(mean: float = 0, stdev: float = 1)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning the Gaussian Distribution 27 28.

\[\text{Gaussian}(x;\mu,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\big( \frac{x-\mu}{\sigma}\big)^2}\]

Parameters

• mean (float) -- mean of the distribution (\(\mu\))

• std (float) -- standard deviation (\(\sigma\)) of the distribution where std > 0

• x (float) -- random variable

References

25

Wikipedia contributors. (2020, December 19). Gaussian distribution. https://en.wikipedia.org/w/index.php?title=Gaussian_distribution&oldid=995237372

26

Weisstein, Eric W. "Gaussian Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianDistribution.html

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Gaussian distribution

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Gaussian distribution

mean() → float

Returns: Mean of the Gaussian distribution

median() → float

Returns: Median of the Gaussian distribution

mode() → float

Returns: Mode of the Gaussian distribution

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Gaussian distribution

std() → float

Returns: Standard deviation of the Gaussian distribution

summary() → Dict[str, Union[float, int, str]]

Returns

Dictionary of Gaussian distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Gaussian distribution

class source.univariate.Infinite.HyperbolicSecant(loc: float, scale: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Hyperbolic Secant 29 30 .

\[\text{HyperbolicSecant}(x; \mu, \sigma) = \frac{\text{sech} \Big( \frac{\pi (x - \mu)}{2 \sigma} \Big)}{2 \sigma}\]

Parameters

• loc (float) -- location parameter \(\mu\)

• scale (float) -- scale parameter \(\sigma > 0\)

• x (float) -- random variable

Referneces:

27

Seigrist, K. (n.d.) The Hyperbolic Secant Distribution. https://www.randomservices.org/random/special/HyperbolicSecant.html.

28

Wolfram Alpha(2021). Hyperbolic Secant. https://www.wolframalpha.com/input/?i=hyberbolic+secant+distribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

kurtosis() → float

Returns: Kurtosis of Hyperbolic Secant Distribution.

mean() → float

Returns: Mean of Hyperbolic Secant Distribution.

mode() → float

Returns: Mode of Hyperbolic Secant Distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of Hyperbolic Secant Distribution.

std() → float

Returns: Standard Deviation of Hyperbolic Secant Distribution.

summary() → Dict[str, float]

Returns

Dictionary of Hyperbolic Secant distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of Hyperbolic Secant Distribution.

class source.univariate.Infinite.Infinite

Bases: univariate._base.Base

Description:

Base class for probability tags.

class source.univariate.Infinite.JohnsonSU(gamma: float, xi: float, delta: float, lmbda: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Generalized Normal Distribution V1 31.

\[\text{JohnsonSU}(x;\gamma,\xi,\delta,\lambda) = \frac{\delta}{\lambda \sqrt{2\pi} \cdot \sqrt{1+\Big(\frac{x-\xi}{\lambda}\Big)}} e^{-0.5\Big(\gamma + \delta \sinh^-1 \Big(\frac{x-\xi}{\lambda}\Big) \Big)^2}\]

Parameters

• gamma (float) -- gamma parameter \(\gamma\)

• xi (float) -- xi parameter \(\xi\)

• delta (float) -- delta parameter \(\delta > 0\)

• lamlda (float) -- lambda parameter \(\lambda > 0\)

• x (float) -- random variable

Reference:

29

Wikipedia Contributors (2021). Johnson's SU-distribution. https://en.wikipedia.org/wiki/Johnson.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

mean() → float

Returns: Variance of the JohnsonSU distribution.

median() → float

Returns: Variance of the JohnsonSU distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

std() → float

Returns: Variance of the JohnsonSU distribution.

summary() → Dict[str, Optional[float]]

Returns

Dictionary of Assymetric Laplace distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the JohnsonSU distribution.

class source.univariate.Infinite.Landau(scale: float, loc: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Generalized Normal Distribution V1 32.

Parameters

• scale (float) -- scale parameter \(c > 0\)

• loc (float) -- location parameter \(\mu\)

• x (float) -- random variable

cdf()

kurtosis() → str

Returns: Variance of the Landau distribution.

mean() → str

Returns: Variance of the Landau distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

skewness() → str

Returns: Variance of the Landau distribution.

std() → str

Returns: Variance of the Landau distribution.

summary() → Dict[str, str]

Returns

Dictionary of Assymetric Landau distirbution moments. This includes standard deviation.

var() → str

Returns: Variance of the Landau distribution.

class source.univariate.Infinite.Laplace(loc: float, scale: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning Laplace Distirbution 33 34.

\[\text{Laplace}(x;\mu, b) = \frac{1}{2b} \exp{- \frac{|x - \mu |}{b}}\]

Parameters

• loc (float) -- loc parameter (\(\mu\))

• scale (float) -- scale parameter (\(b > 0\))

• x (float) -- random variable

Reference:

30

Wikipedia contributors. (2020, December 21). Laplace distribution. https://en.wikipedia.org/w/index.php?title=Laplace_distribution&oldid=995563221

31

Wolfram Research (2007), LaplaceDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceDistribution.html (updated 2016).

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Laplace distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Laplace distribution.

mean() → float

Returns: Mean of the Laplace distribution.

median() → float

Returns: Median of the Laplace distribution.

mode() → float

Returns: Mode of the Laplace distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Laplace distribution.

std() → float

Returns: Standard deviation of the Laplace distribution

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Laplace distirbution moments. This includes standard deviation.

var() → Union[int, float]

Returns: Variance of the Laplace distribution.

class source.univariate.Infinite.Logistic(location: float, scale: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning Logistic Distirbution 35 36.

\[\text{Logistic}(x;\mu,s) = \frac{\exp{(-(x-\mu)/s)}} {s(1+\exp(-(x-\mu)/s)^2)}\]

Parameters

• location (float) -- location parameter (\(\mu\))

• scale (float) -- scale parameter (\(s\)) x > 0

• x (float) -- random variable

Reference:

32

Wikipedia contributors. (2020, December 12). Logistic distribution. https://en.wikipedia.org/w/index.php?title=Logistic_distribution&oldid=993793195

33

Wolfram Research (2007), LogisticDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LogisticDistribution.html (updated 2016).

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Logistic distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Logistic distribution.

mean() → float

Returns: Mean of the Logistic distribution.

median() → float

Returns: Median of the Logistic distribution.

mode() → float

Returns: Mode of the Logistic distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Logistic distribution.

std() → float

Returns: Standard deviation of the Logistic distribution.

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Logistic distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Logistic distribution.

class source.univariate.Infinite.SkewNormal(loc: float, scale: float, shape: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Generalized Normal Distribution V1 37 38.

\[\text{SkewNormal}(x;\xi,\omega,\alpha) = \frac{e^{\frac{-(x-\xi)^2}{2 \omega^2} \text{erfc} \Big( - \frac{a(x-\xi)}{\sqrt{2} \omega} \Big) }}{ \sqrt{2\pi} \omega}\]

Parameters

• loc (float) -- location parameter \(\xi\)

• scale (float) -- scale parameter \(\omega\)

• shape (float) -- shape parameter \(\alpha\)

• x (float) -- random variable

Reference:

34

Wikipedia Contributors (2021). Skew Normal Distribution. https://en.wikipedia.org/wiki/Skew_normal_distribution.

35

Wolfram Research (2010), SkewNormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SkewNormalDistribution.html (updated 2016).

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

kurtosis() → float

Returns: Kurtosis of the Skew Normal distribution.

mean() → float

Returns: Mean of the Skew Normal distribution.

median() → float

Default implementation of the median. Returns NotImplemented.

mode() → float

Default implementation of the mode. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variables

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Skew Normal distribution.

std() → float

Returns: Standard Deviation of the Skew Normal distribution.

summary() → Dict[str, Optional[float]]

Returns

Dictionary of Assymetric Skew Normal distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Skew Normal distribution.

class source.univariate.Infinite.Slash(loc: float, scale: float, shape: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Slash Distribution 39.

\[ext{Slash}(x) = {\displaystyle {egin{cases}{\]

rac { arphi (0)- arphi (x)}{x^{2}}}&x eq 0{ rac {1}{2{sqrt {2pi }}}}&x=0\end{cases}}}

Args:

x (float): random variable

Reference:

36

Wikipedia Contributors (2021). Slash Distribution. https://en.wikipedia.org/wiki/Slash_distribution

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

entropy() → float

Default implementation of entropy. Returns NotImplemented.

kurtosis() → str

Returns: Kurtosis of the Slash distribution.

mean() → str

Returns: Mean of the Slash distribution.

median() → float

Returns: Median of the Slash distribution.

mode() → float

Returns: Mode of the Slash distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

skewness() → str

Returns: Skewness of the Slash distribution.

std() → str

Returns: Standard Deviation of the Slash distribution.

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Slash distirbution moments. This includes standard deviation.

var() → str

Returns: Variance of the Slash distribution.

class source.univariate.Infinite.T(df: int)

Bases: source.univariate.Infinite.Infinite

This class contains implementation of the Student's Distribution for calculating the probablity density function and cumulative distribution function. Additionally, a t-table __generator is also provided by p-value method. Note that the implementation of T(Student's) distribution is defined by beta-functions 40 41.

\[\text{T}(x;\nu) = \frac{1}{\sqrt{\nu}\text{B}\Big(\frac{1}{2}, \frac{\nu}{2}\Big)} \Big(1 + \frac{t^2}{\nu}\Big) ^{-\frac{\nu+1}{2}}\]

\[\begin{split}\text{T}(x;\nu) = \frac{\Gamma\Big( \frac{\nu+1}{2} \Big)}{\sqrt{\nu\pi} \Gamma{\Big( \frac{\nu}{2} \Big)}} \Big(1 + \frac{x^2}{\nu}\Big) ^{- \frac{v+1}{2}} \\\end{split}\]

Parameters

• df (int) -- degrees of freedom (\(\nu\)) where df > 0

• x (float) -- random variable

References

37

Kruschke JK (2015). Doing Bayesian Data Analysis (2nd ed.). Academic Press. ISBN 9780124058880. OCLC 959632184.

38

Weisstein, Eric W. "Student's t-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Studentst-Distribution.html

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of T-distribution

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → Union[float, str]

Returns: Kurtosis of the T-distribution

mean() → Union[float, str]

Mean of the T-distribution. :returns: 0 for df > 1, otherwise Indeterminate.

median() → float

Returns: Median of the T-distribution

mode() → float

Returns: Mode of the T-distribution

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → Union[float, str]

Returns: Skewness of the T-distribution

std() → Union[float, str]

Returns: Standard Deviation of the T-distribution

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of T distirbution moments. This includes standard deviation.

var() → Union[float, str]

Returns: Variance of the T-distribution

class source.univariate.Infinite.VarianceGamma(loc: float, scale: float, shape: float)

Bases: source.univariate.Infinite.Infinite

This class contains methods concerning to Generalized Normal Distribution V1 42.

Parameters

• loc (float) -- location parameter \(\mu\)

• scale (float) -- scale parameter \(\alpha\)

• shape (float) -- shape parameter \(\beta\)

• x (float) -- random variable

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

entropy() → float

Default implementation of entropy. Returns NotImplemented.

kurtosis() → float

Default implementation of kurtosis. Returns NotImplemented.

mean() → float

Default implementation of the mean. Returns NotImplemented.

median() → float

Default implementation of the median. Returns NotImplemented.

mode() → float

Default implementation of the mode. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

skewness() → float

Default implementation of skewness. Returns NotImplemented.

std() → float

Default implementation of the standard deviation. Returns NotImplemented.

summary() → Dict[str, Optional[float]]

Returns

Dictionary of Assymetric Laplace distirbution moments. This includes standard deviation.

var() → float

Default implementation of the variance. Returns NotImplemented.

source.univariate.SemiInfinite module

class source.univariate.SemiInfinite.Benini(shape_a: float, shape_b: float, scale: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning the Benini Distribution 43 44.

\[\text{Benini}(x;\alpha,\beta,\sigma) = e^{\alpha \log(x/\alpha) - \beta[\log(x/\sigma)]^2} \Big(\frac{\alpha}{x} + \frac{2 \beta \log(x/\sigma)}{x} \Big)\]

Parameters

• shape_a (float) -- shape parameter \(\alpha > 0\)

• shape_b (float) -- shape parameter \(\beta > 0\)

• scale (float) -- scale parameter \(\sigma > 0\)

• x (float) -- random variables

References

39

Wikipedia Contributors (2021). Benini Distribution. https://en.wikipedia.org/wiki/Benini_distribution.

40

Wolfram Research (2010), BeniniDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BeniniDistribution.html (updated 2016).

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) or interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

mean() → float

Default implementation of the mean. Returns NotImplemented.

median() → float

Default implementation of the median. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x <= sigma

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

std() → float

Default implementation of the standard deviation. Returns NotImplemented.

summary() → Dict[str, float]

Returns

Dictionary of Benini distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of Benini distribution Reference: Wolfram (2021). BeniniDistribution. https://reference.wolfram.com/language/ref/BeniniDistribution.html.

class source.univariate.SemiInfinite.BetaPrime(alpha: float, beta: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning Beta prime Distirbution 45 .

\[\text{BetaPrime}(x;\alpha,\beta) = \frac{x^{\alpha -1}(1+x)^{-\alpha -\beta}}{\text{B}(\alpha ,\beta )}\]

Parameters

• alpha (float) -- shape parameter where alpha > 0

• beta (float) -- shape parameter where beta > 0

• x (float) -- random variable where x >= 0

Reference:

41

Wikipedia contributors. (2020, October 8). Beta prime distribution. https://en.wikipedia.org/w/index.php?title=Beta_prime_distribution&oldid=982458594

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Raises

ValueError -- when there exist a value of x less than 0

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy()

Returns: differential entropy of the Beta prime distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → str

Returns: Kurtosis of the Beta prime distribution.

mean() → Union[float, str]

Returns: Mean of the Beta prime distribution.

median() → str

Returns: Median of the Beta prime distribution.

mode() → float

Returns: Mode of the Beta prime distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x less than 0

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → Union[float, str]

Returns: Skewness of the Beta prime distribution.

std() → Union[float, str]

Returns: Standard deviation of the Log logistic distribution

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of BetaPrime distirbution moments. This includes standard deviation.

var() → Union[float, str]

Returns: Variance of the Beta prime distribution.

class source.univariate.SemiInfinite.BurrXII(c: float, k: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

kurtosis() → float

Default implementation of kurtosis. Returns NotImplemented.

mean() → float

Default implementation of the mean. Returns NotImplemented.

median() → float

Default implementation of the median. Returns NotImplemented.

mode() → float

Default implementation of the mode. Returns NotImplemented.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x <= 0

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Default implementation of skewness. Returns NotImplemented.

summary() → Dict[str, float]

Returns

Dictionary of Burr distirbution moments. This includes standard deviation.

var() → float

Default implementation of the variance. Returns NotImplemented.

class source.univariate.SemiInfinite.Chi(df: int)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning the Chi distribution 46 47.

\[\text{Chi}(x; k) = \frac{2^{1-k/2} e^{-\frac{x^2}{2}} x^{k-1}}{\Gamma \Big(\frac{k}{2} \Big)}\]

Parameters

• df (int) -- degrees of freedom where df > 0

• x (float) -- random variable

References

42

Weisstein, Eric W. "Chi Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChiDistribution.html

43

Wikipedia contributors. (2020, October 16). Chi distribution. https://en.wikipedia.org/w/index.php?title=Chi_distribution&oldid=983750392

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of Chi distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Chi distribution.

mean() → float

Returns: Mean of the Chi distribution.

median() → Union[float, int]

Returns: Median of the Chi distribution.

mode() → Union[float, str]

Returns: Mode of the Chi distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], np.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x <= 0

Returns

evaluation of pdf at x

Return type

Union[float, np.ndarray]

skewness() → Union[float, int]

Returns: Skewness of the Chi distribution.

std() → Union[float, int]

Returns: Standard deviation of the Chi distribution.

summary() → Dict[str, Union[float, int, str]]

Returns

Dictionary of Chi distirbution moments. This includes standard deviation.

var() → Union[float, int]

Returns: Variance of the Chi distribution.

class source.univariate.SemiInfinite.ChiSquare(df: int)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning the Chi-square distribution 48 49 50.

\[\text{ChiSquare}(x;k) = \frac{2^{-k/2} e^{-x/2} x^{k/2 - 1}}{\Gamma\Big(\frac{k}{2} \Big)}\]

Parameters

• df (int) -- degrees of freedom (\(k\)) where df > 0

• x (float) -- random variable.

References

44

Wolfram Alpha (2020). Chi-squared distribution. https://www.wolframalpha.com/input/?i=chi+squared+distribution.

45

Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Chi-SquaredDistribution.html

46

Wikipedia contributors. (2020, December 13). Chi-square distribution. https://en.wikipedia.org/w/index.php?title=Chi-square_distribution&oldid=994056539

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Raises

ValueError -- when there exist a value of x less than 0

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of Chi-square distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Chi-square distribution.

mean() → float

Returns: Mean of the Chi-square distribution.

median() → float

Returns: Median of the Chi-square distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a vaue less than 0

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Chi-square distribution.

std() → float

Returns: Standard deviation of the Chi-square distribution.

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Chi-Square distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Chi-square distribution.

class source.univariate.SemiInfinite.Dagum(shape_p: float, shape_a: float, scale: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning Dagum Distribution 51.

\[\text{Dagum}(x;a,p,b) = \frac{ap}{x} \Bigg(\frac{\frac{x}{b}^{ap}}{\Big(\frac{x}{b}^a + 1\Big)^{p+1}}\Bigg)\]

Parameters

• shape_p (float) -- shape parameter \(p > 0\)

• shape_a (float) -- shape parameter \(a > 0\)

• scale (float) -- scale parameter \(b > 0\)

• x (float) -- random variables

References

47

Wikipedia Contributors (2021). Dagum Distribution. https://en.wikipedia.org/wiki/Dagum_distribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

mean() → Union[float, str]

Returns: Mean of the Dagum distribution.

median() → float

Returns: Median of the Dagum distribution.

mode() → float

Returns: Mode of the Dagum distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x <= 0

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

std() → Union[float, str]

Returns: Standard Deviation of the Dagum distribution.

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Dagum distirbution moments. This includes standard deviation.

var() → Union[float, str]

Returns: Variance of the Dagum distribution.

class source.univariate.SemiInfinite.Erlang(shape: int, rate: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning Erlang Distirbution 52 53 54.

\[\text{Erlang}(x; k, \lambda) = \frac{\lambda^{k} x^{k-1} e^{-\lambda x}}{\Gamma(k)}\]

Parameters

• shape (int) -- shape parameter (\(k\)) where shape > 0

• rate (float) -- rate parameter (\(\lambda\)) where rate >= 0

• x (float) -- random variable where x >= 0

Reference:

48

Wolfram Alpha (2021). Erlang Distribution. https://www.wolframalpha.com/input/?i=erlang+distribution.

49

Wikipedia contributors. (2021, January 6). Erlang distribution. https://en.wikipedia.org/w/index.php?title=Erlang_distribution&oldid=998655107

50

Weisstein, Eric W. "Erlang Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ErlangDistribution.html

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Raises

ValueError -- when there exist a data value of x that is less than 0 or greater than 1

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Erlang distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Erlang distribution.

mean() → float

Returns: Mean of the Erlang distribution.

median() → str

Returns: Median of the Erlang distribution.

mode() → Union[float, str]

Returns: Mode of the Erlang distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x that is less than 0 or greater than 1

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Erlang distribution.

std() → float

Returns: Standard deviation of the Eerlang distribution.

summary() → Dict[str, Union[float, int, str]]

Returns

Dictionary of Erlang distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Erlang distribution.

class source.univariate.SemiInfinite.Exponential(rate: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contans methods for evaluating Exponential Distirbution 55 56 57.

\[\text{Exponential}(x;\lambda) = \lambda e^{-\lambda x}\]

Parameters

• rate (-) -- rate parameter (\(\lambda\)) where rate > 0

• x (-) -- random variable where x > 0

References

51

Wolfram Alpha (2021). Exponential Distirbution. https://www.wolframalpha.com/input/?i=exponential+distribution.

52

Weisstein, Eric W. "Exponential Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialDistribution.html

53

Wikipedia contributors. (2020, December 17). Exponential distribution. https://en.wikipedia.org/w/index.php?title=Exponential_distribution&oldid=994779060

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entorpy() → float

Returns: differential entropy of the Exponential distribution

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Exponential distribution

mean() → float

Returns: Mean of the Exponential distribution

median() → float

Returns: Median of the Exponential distribution

mode() → float

Returns: Mode of the Exponential distribution

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Exponential distribution

std() → float

Returns: Standard deviation of the Exponential distribution

summary() → Dict[str, float]

Returns

Dictionary of Exponential distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Exponential distribution

class source.univariate.SemiInfinite.F(df1: int, df2: int)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning the F-distribution 58 59 60.

\[\text{F}(x;d_1, d_2) = \frac{1}{\text{B}(d_1/2,d_2/2)} \Big( \frac{d_1}{d_2} \Big)^{d_1/2} x^{d_1/2 - 1} \Big(1 + \frac{d_1}{d_2}x\Big) ^{-(d_1+d_2)/2}\]

Parameters

• df1 (int) -- first degrees of freedom where df1 > 0

• df2 (int) -- second degrees of freedom where df2 > 0

• x (float) -- random variable where x > 0

References

54

Mood, Alexander; Franklin A. Graybill; Duane C. Boes (1974). Introduction to the Theory of Statistics (Third ed.). McGraw-Hill. pp. 246–249. ISBN 0-07-042864-6.

55

Weisstein, Eric W. "F-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/F-Distribution.html

56

NIST SemaTech (n.d.). F-Distribution. Retrived from https://www.itl.nist.gov/div898/handbook/eda/section3/eda3665.htm

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s)

Returns

evaluates cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → Union[float, int]

Returns: differential entropy of F-distribution.

Reference: Lazo, A.V.; Rathie, P. (1978). "On the entropy of continuous probability distributions". IEEE Transactions on Information Theory

mean() → Union[float, str]

Returns: Mean of the F-distribution.

mode() → Union[float, str]

Returns: Mode of the F-distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value such that x < 0

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → Union[float, str]

Returns: Skewness of the F-distribution.

std() → Union[float, str]

Returns: Standard deviation of the F-distribution.

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of F distirbution moments. This includes standard deviation.

var() → Union[float, str]

Returns: Variance of the F-distribution.

class source.univariate.SemiInfinite.Gamma(shape: float, b: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning a variant of Gamma distribution 61.

\[\text{Gamma}(x;a,b) = \frac{1}{b^a \Gamma(a)} \ x^{a-1} e^{\frac{-x}{b}}\]

Parameters

• shape (float) -- shape parameter (\(a\)) where shape > 0

• scale (float) -- scale parameter (\(b\)) where scale > 0

• x (float) -- random variable where x > 0

References

57

Matlab(2020). Gamma Distribution. https://www.mathworks.com/help/stats/gamma-distribution.html

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Gamma distribution

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Gamma distribution

mean() → Union[float, int]

Returns: Mean of the Gamma distribution

median() → str

Returns: Median of the Gamma distribution.

mode() → Union[float, int]

Returns: Mode of the Gamma distribution

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x that is less than 0

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Gamma distribution

std() → float

Returns: Standard deviation of the Gamma distribution

summary() → Dict[str, Union[float, int, str]]

Returns

Dictionary of Gamma distirbution moments. This includes standard deviation.

var() → Union[float, int]

Returns: Variance of the Gamma distribution

class source.univariate.SemiInfinite.Gumbell(location: float, scale: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning Gumbel Distirbution 62.

\[\text{Gumbel}(x;\mu,\beta) = \frac{1}{\beta} \exp{-\Big( \frac{x-\mu}{\beta} + \exp{ \frac{x-\mu}{\beta}} \Big)}\]

Parameters

• location (float) -- location parameter (\(\mu\))

• scale (float) -- scale parameter (\(\beta\)) where scale > 0

• x (float) -- random variable

Reference:

58

Wikipedia contributors. (2020, November 26). Gumbel distribution. https://en.wikipedia.org/w/index.php?title=Gumbel_distribution&oldid=990718796

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

kurtosis() → float

Returns: Kurtosis of the Gumbel distribution.

mean() → float

Returns: Mean of the Gumbel distribution.

median() → float

Returns: Median of the Gumbel distribution.

mode() → float

Returns: Mode of the Gumbel distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Gumbel distribution.

std() → float

Returns: Standard deviation of the Gumbel distribution.

summary() → Dict[str, float]

Returns

Dictionary of Gumbel distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Gumbel distribution.

class source.univariate.SemiInfinite.LogNormal(mean: float, std: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning the Log Normal Distribution 63 64.

\[\text{LogNormal}(x;\mu,\sigma) = \frac{1}{x\sigma\sqrt{2\pi}} \exp{\Big( - \frac{(\ln x - \mu)^2}{2\sigma^2} \Big)}\]

Parameters

• mean (float) -- mean parameter (\(\mu\))

• std (float) -- standard deviation (\(\sigma\)) where std > 0

• x (float) -- random variable where x >= 0

References

59

Weisstein, Eric W. "Log Normal Distribution." From MathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/LogNormalDistribution.html

60

Wikipedia contributors. (2020, December 18). Log-normal distribution. https://en.wikipedia.org/w/index.php?title=Log-normal_distribution&oldid=994919804

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the log normal distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the log normal distribution.

mean() → float

Returns: Mean of the log normal distribution.

median() → float

Returns: Median of the log normal distribution.

mode() → float

Returns: Mode of the log normal distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x < 0

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the log normal distribution.

std() → float

Returns: Standard deviation of the log normal distribution

summary() → Dict[str, Union[float, int, str]]

Returns

Dictionary of Log Normal distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the log normal distribution.

class source.univariate.SemiInfinite.MaxwellBoltzmann(a: int)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning Maxwell-Boltzmann Distirbution 65 66.

\[\text{MaxwellBoltzmann}(x;a) = \sqrt{\frac{2}{\pi}} \frac{x^2 \exp{-x^2/(2a^2)}}{a^3}\]

Parameters

• a (int) -- parameter where a > 0

• x (float) -- random variable where x >= 0

Reference:

61

Wikipedia contributors. (2021, January 12). Maxwell–Boltzmann distribution. https://en.wikipedia.org/w/index.php?title=Maxwell%E2%80%93Boltzmann_distribution&oldid=999883013

62

Wolfram Alpha (2021). Maxwell Distribution. https://www.wolframalpha.com/input/?i=maxwelldistribution.

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) or interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Maxwell-Boltzmann distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Maxwell-Boltzmann distribution.

mean() → float

Returns: Mean of the Maxwell-Boltzmann distribution.

median() → Union[float, str]

Returns: Median of the Maxwell-Boltzmann distribution.

mode() → float

Returns: Mode of the Maxwell-Boltzmann distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x less than 0

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Maxwell-Boltzmann distribution.

std() → float

Returns: Standard deviation of the Maxwell-Boltzmann distribution

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Maxwell-Boltzmann distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Maxwell-Boltzmann distribution.

class source.univariate.SemiInfinite.Pareto(shape: float, scale: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning the Pareto Distribution Type 1 67 68.

\[\text{Pareto}(x;x_m, a) = \frac{a x_m^a}{x^{a+1}}\]

Parameters

• scale (float) -- scale parameter (\(x_m\)) where scale > 0

• shape (float) -- shape parameter (\(a\)) where shape > 0

• x (float) -- random variable where shape <= x

References

63

Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 978-0-89974-012-6.

64

Wikipedia contributors. (2020, December 1). Pareto distribution. https://en.wikipedia.org/w/index.php?title=Pareto_distribution&oldid=991727349

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Pareto distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → Union[float, str]

Returns: Kurtosis of the Pareto distribution.

mean() → float

Returns: Mean of the Pareto distribution.

median() → float

Returns: Median of the Pareto distribution.

mode() → float

Returns: Mode of the Pareto distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there is a case that a random variable is greater than the value of shape parameter

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → Union[float, str]

Returns: Skewness of the Pareto distribution.

std() → float

Returns: Variance of the Pareto distribution

summary() → Dict[str, Union[float, str]]

Returns

Dictionary of Pareto distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Pareto distribution.

class source.univariate.SemiInfinite.Rayleigh(scale: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning Rayleigh Distirbution 69 70 [#]_.

\[\text{Rayleigh}(x;\sigma) = \frac{x}{\sigma^2} \exp{-(x^2/(2\sigma^2))}\]

Parameters

• scale (float) -- scale parameter (\(\sigma\)) where scale > 0

• x (float) -- random variable where x >= 0

Reference:

65

Wolfram Alpha (2021). Rayleigh distribution. https://www.wolframalpha.com/input/?i=rayleigh+distribution.

66

Wikipedia contributors. (2020, December 30). Rayleigh distribution. https://en.wikipedia.org/w/index.php?title=Rayleigh_distribution&oldid=997166230

67

Weisstein, Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RayleighDistribution.html

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Rayleigh distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

kurtosis() → float

Returns: Kurtosis of the Rayleigh distribution.

mean() → float

Returns: Mean of the Rayleigh distribution.

median() → float

Returns: Median of the Rayleigh distribution.

mode() → float

Returns: Mode of the Rayleigh distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a value of x that is less than 0

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Rayleigh distribution.

std() → float

Returns: Standard deviation of the Rayleigh distribution

summary() → Dict[str, Union[float, Tuple[float]]]

Returns

Dictionary of Rayleigh distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Rayleigh distribution.

class source.univariate.SemiInfinite.SemiInfinite

Bases: univariate._base.Base

Description:

Base class for probability tags.

class source.univariate.SemiInfinite.Weibull(shape: float, scale: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning Weibull Distirbution [#]_ [#]_.

\[\text{Weibull}(x;\lambda, k) = \frac{k}{\lambda} \Big( \frac{x}{\lambda}\Big)^{k-1} \exp(-(x/\lambda)^k)\]

Parameters

• shape (float) -- shape parameter (\(\lambda\)) where shape >= 0

• scale (float) -- scale parameter (\(k\)) where scale >= 0

• randvar (float) -- random variable where x >= 0

Reference:

68

Wolfram Alpha (2021). Weibull Distribution. https://www.wolframalpha.com/input/?i=weibul+distribution.

69

Wikipedia contributors. (2020, December 13). Weibull distribution. https://en.wikipedia.org/w/index.php?title=Weibull_distribution&oldid=993879185

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of cdf at x

Return type

Union[float, numpy.ndarray]

entropy() → float

Returns: differential entropy of the Weilbull distribution.

Reference: Park, S.Y. & Bera, A.K.(2009). Maximum entropy autoregressive conditional heteroskedasticity model. Elsivier. link: http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf

mean() → float

Returns: Mean of the Weibull distribution.

median() → float

Returns: Median of the Weibull distribution.

mode() → float

Returns: Mode of the Weibull distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → float

Returns: Skewness of the Weilbull distribution

std() → float

Returns: Standard deviation of the Weilbull distribution

summary() → Dict[str, Union[float, int, str]]

Returns

Dictionary of Weibull distirbution moments. This includes standard deviation.

var() → float

Returns: Variance of the Weibull distribution.

class source.univariate.SemiInfinite.WeibullInverse(shape: float, scale: float, loc: float)

Bases: source.univariate.SemiInfinite.SemiInfinite

This class contains methods concerning inverse Weilbull or the Fréchet Distirbution [#]_.

\[\text{WeibullInverse}(x;a,s,m) = \frac{a}{s} \Big(\frac{x-m}{s} \Big) ^{-1-a} \exp{\Big(-\frac{x-m}{s} \Big)^{-a}}\]

Parameters

• shape (float) -- shape parameter (\(a\)) where shape >= 0

• scale (float) -- scale parameter (\(s\)) where scale >= 0

• loc (float) -- loc parameter (\(m\))

• randvar (float) -- random variable where x > loc

Reference:

70

Wikipedia contributors. (2020, December 7). Fréchet distribution. https://en.wikipedia.org/w/index.php?title=Fr%C3%A9chet_distribution&oldid=992938143

cdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- data point(s) of interest

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

kurtosis() → Union[float, str]

Returns: Kurtosis of the Fréchet distribution.

mean() → float

Returns: Mean of the Fréchet distribution.

median() → float

Returns: Median of the Fréchet distribution.

mode() → float

Returns: Mode of the Fréchet distribution.

pdf(x: Union[List[float], numpy.ndarray, float]) → Union[float, numpy.ndarray]

Parameters

x (Union[List[float], numpy.ndarray, float]) -- random variable(s)

Raises

ValueError -- when there exist a random variate less than or equal to loc parameter

Returns

evaluation of pdf at x

Return type

Union[float, numpy.ndarray]

skewness() → Union[float, str]

Returns: Skewness of the Fréchet distribution.

std() → Union[float, str]

Returns: Standard devtiation of the Fréchet distribution.

summary() → Dict[str, Union[float, int, str]]

Returns

Dictionary of Fréchet distirbution moments. This includes standard deviation.

var() → Union[float, str]

Returns: Variance of the Fréchet distribution.

Module contents

We want to avoid name collisions and keep our names in the module they are defined in. There are two ways we can deal about it: - using __all__ = [DO_NOT_WILD_IMPORT] to avoid from [module] import * - specify which class you are importing e.g. `from .Base import Base