Mathematical and statistical functions for the Normal distribution, which is commonly used in significance testing, for representing models with a bell curve, and as a result of the central limit theorem.

Value

Returns an R6 object inheriting from class SDistribution.

Details

The Normal distribution parameterised with variance, \(\sigma^2\), and mean, \(\mu\), is defined by the pdf, $$f(x) = exp(-(x-\mu)^2/(2\sigma^2)) / \sqrt{2\pi\sigma^2}$$ for \(\mu \epsilon R\) and \(\sigma^2 > 0\).

Distribution support

The distribution is supported on the Reals.

Default Parameterisation

Norm(mean = 0, var = 1)

Omitted Methods

N/A

Also known as

Also known as the Gaussian distribution.

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

See also

Super classes

distr6::Distribution -> distr6::SDistribution -> Normal

Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

packages

Packages required to be installed in order to construct the distribution.

Methods

Public methods

Inherited methods

Method new()

Creates a new instance of this R6 class.

Usage

Normal$new(mean = NULL, var = NULL, sd = NULL, prec = NULL, decorators = NULL)

Arguments

mean

(numeric(1))
Mean of the distribution, defined on the Reals.

var

(numeric(1))
Variance of the distribution, defined on the positive Reals.

sd

(numeric(1))
Standard deviation of the distribution, defined on the positive Reals. sd = sqrt(var). If provided then var ignored.

prec

(numeric(1))
Precision of the distribution, defined on the positive Reals. prec = 1/var. If provided then var ignored.

decorators

(character())
Decorators to add to the distribution during construction.


Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation $$E_X(X) = \sum p_X(x)*x$$ with an integration analogue for continuous distributions.

Usage

Normal$mean(...)

Arguments

...

Unused.


Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage

Normal$mode(which = "all")

Arguments

which

(character(1) | numeric(1)
Ignored if distribution is unimodal. Otherwise "all" returns all modes, otherwise specifies which mode to return.


Method variance()

The variance of a distribution is defined by the formula $$var_X = E[X^2] - E[X]^2$$ where \(E_X\) is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage

Normal$variance(...)

Arguments

...

Unused.


Method skewness()

The skewness of a distribution is defined by the third standardised moment, $$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution.

Usage

Normal$skewness(...)

Arguments

...

Unused.


Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment, $$k_X = E_X[\frac{x - \mu}{\sigma}^4]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

Usage

Normal$kurtosis(excess = TRUE, ...)

Arguments

excess

(logical(1))
If TRUE (default) excess kurtosis returned.

...

Unused.


Method entropy()

The entropy of a (discrete) distribution is defined by $$- \sum (f_X)log(f_X)$$ where \(f_X\) is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage

Normal$entropy(base = 2, ...)

Arguments

base

(integer(1))
Base of the entropy logarithm, default = 2 (Shannon entropy)

...

Unused.


Method mgf()

The moment generating function is defined by $$mgf_X(t) = E_X[exp(xt)]$$ where X is the distribution and \(E_X\) is the expectation of the distribution X.

Usage

Normal$mgf(t, ...)

Arguments

t

(integer(1))
t integer to evaluate function at.

...

Unused.


Method cf()

The characteristic function is defined by $$cf_X(t) = E_X[exp(xti)]$$ where X is the distribution and \(E_X\) is the expectation of the distribution X.

Usage

Normal$cf(t, ...)

Arguments

t

(integer(1))
t integer to evaluate function at.

...

Unused.


Method pgf()

The probability generating function is defined by $$pgf_X(z) = E_X[exp(z^x)]$$ where X is the distribution and \(E_X\) is the expectation of the distribution X.

Usage

Normal$pgf(z, ...)

Arguments

z

(integer(1))
z integer to evaluate probability generating function at.

...

Unused.


Method clone()

The objects of this class are cloneable with this method.

Usage

Normal$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.