# ChiSquare: Create a Chi-Square distribution In distributions3: Probability Distributions as S3 Objects

## Description

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

## Usage

 `1` ```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 χ^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) = 1 / (2 π σ^2) exp(-(x - μ)^2 / (2 σ^2))

Cumulative distribution function (c.d.f):

The cumulative distribution function has the form

F(t) = integral_{-∞}^t 1 / (2 π σ^2) exp(-(x - μ)^2 / (2 σ^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 Φ(t) also stands for the c.d.f. of a standard normal evaluated at t. Z-tables list the value of Φ(t) for various t.

Moment generating function (m.g.f):

E(e^(tX)) = e^(μ t + σ^2 t^2 / 2)

## Value

A `ChiSquare` object.

## Transformations

A squared standard `Normal()` distribution is equivalent to a χ^2_1 distribution with one degree of freedom. The χ^2 distribution is a special case of the `Gamma()` distribution with shape (TODO: check this) parameter equal to a half. Sums of χ^2 distributions are also distributed as χ^2 distributions, where the degrees of freedom of the contributing distributions get summed. The ratio of two χ^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`, `Exponential`, `FisherF`, `Gamma`, `LogNormal`, `Logistic`, `Normal`, `StudentsT`, `Tukey`, `Uniform`, `Weibull`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```set.seed(27) X <- ChiSquare(5) 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)) ```

distributions3 documentation built on Sept. 3, 2019, 5:06 p.m.