distributional (version 0.5.0)

dist_chisq: The (non-central) Chi-Squared 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

dist_chisq(df, ncp = 0)

Arguments

df

degrees of freedom (non-negative, but can be non-integer).

ncp

non-centrality parameter (non-negative).

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

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):

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

See Also

stats::Chisquare

Examples

dist <- dist_chisq(df = c(1,2,3,4,6,9))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)