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

ChiSquare(df)

## Arguments

df Degrees of freedom. Must be positive.

## Value

A ChiSquare object.

## 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):

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

## 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, Exponential, FisherF, Gamma, LogNormal, Logistic, Normal, StudentsT, Tukey, Uniform, Weibull

## Examples


set.seed(27)

X <- ChiSquare(5)
X#> Chi Square distribution (df = 5)
random(X, 10)#>  [1] 11.2129049  7.8935724  2.1298341  5.2084236  5.4563211  3.6636712
#>  [7] 10.9823299  0.7858347  4.8748588  1.7938110
pdf(X, 2)#> [1] 0.1383692log_pdf(X, 2)#> [1] -1.97783
cdf(X, 4)#> [1] 0.450584quantile(X, 0.7)#> [1] 6.06443
cdf(X, quantile(X, 0.7))#> [1] 0.7quantile(X, cdf(X, 7))#> [1] 7