dist_chisq {distributional}

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)