dist_chisq: The (non-central) Chi-Squared Distribution

In distributional: Vectorised Probability Distributions

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-centrality parameter). Can be any positive real number.
ncp	Non-centrality parameter. Can be any non-negative real number. Defaults to 0 (central chi-squared distribution).

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_chisq.html

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

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

Mean: k + \lambda

Variance: 2(k + 2\lambda)

Probability density function (p.d.f):

For the central chi-squared distribution (\lambda = 0):

f(x) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{k/2 - 1} e^{-x/2}

For the non-central chi-squared distribution (\lambda > 0):

f(x) = \frac{1}{2} e^{-(x+\lambda)/2} \left(\frac{x}{\lambda}\right)^{k/4-1/2} I_{k/2-1}\left(\sqrt{\lambda x}\right)

where I_\nu(z) is the modified Bessel function of the first kind.

Cumulative distribution function (c.d.f):

For the central chi-squared distribution (\lambda = 0):

F(x) = \frac{\gamma(k/2, x/2)}{\Gamma(k/2)} = P(k/2, x/2)

where \gamma(s, x) is the lower incomplete gamma function and P(s, x) is the regularized gamma function.

For the non-central chi-squared distribution (\lambda > 0):

F(x) = \sum_{j=0}^{\infty} \frac{e^{-\lambda/2} (\lambda/2)^j}{j!} P(k/2 + j, x/2)

This is approximated numerically.

Moment generating function (m.g.f):

For the central chi-squared distribution (\lambda = 0):

E(e^{tX}) = (1 - 2t)^{-k/2}, \quad t < 1/2

For the non-central chi-squared distribution (\lambda > 0):

E(e^{tX}) = \frac{e^{\lambda t / (1 - 2t)}}{(1 - 2t)^{k/2}}, \quad t < 1/2

Skewness:

\gamma_1 = \frac{2^{3/2}(k + 3\lambda)}{(k + 2\lambda)^{3/2}}

For the central case (\lambda = 0), this simplifies to \sqrt{8/k}.

Excess Kurtosis:

\gamma_2 = \frac{12(k + 4\lambda)}{(k + 2\lambda)^2}

For the central case (\lambda = 0), this simplifies to 12/k.

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)

distributional documentation built on June 11, 2026, 9:07 a.m.