ChiSquare {distributions3} R Documentation

## Create a Chi-Square distribution

### Description

Chi-square distributions show up often in frequentist settings as the sampling distribution of test statistics, especially in maximum likelihood estimation settings.

### Usage

ChiSquare(df)


### Arguments

 df Degrees of freedom. Must be positive.

### Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a \chi^2 random variable with df = k.

Support: R^+, the set of positive real numbers

Mean: k

Variance: 2k

Probability density function (p.d.f):

 f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2} 

Cumulative distribution function (c.d.f):

The cumulative distribution function has the form

 F(t) = \int_{-\infty}^t \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2} dx 

but this integral does not have a closed form solution and must be approximated numerically. The c.d.f. of a standard normal is sometimes called the "error function". The notation \Phi(t) also stands for the c.d.f. of a standard normal evaluated at t. Z-tables list the value of \Phi(t) for various t.

Moment generating function (m.g.f):

 E(e^{tX}) = e^{\mu t + \sigma^2 t^2 / 2} 

### Value

A ChiSquare object.

### Transformations

A squared standard Normal() distribution is equivalent to a \chi^2_1 distribution with one degree of freedom. The \chi^2 distribution is a special case of the Gamma() distribution with shape (TODO: check this) parameter equal to a half. Sums of \chi^2 distributions are also distributed as \chi^2 distributions, where the degrees of freedom of the contributing distributions get summed. The ratio of two \chi^2 distributions is a FisherF() distribution. The ratio of a Normal() and the square root of a scaled ChiSquare() is a StudentsT() distribution.

Other continuous distributions: Beta(), Cauchy(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

### Examples


set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))


[Package distributions3 version 0.2.1 Index]