dist_chisq
The (non-central) Chi-Squared Distribution
Description
![[Stable]](../help/figures/lifecycle-stable.svg)
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
Chi-square distributions show up often in frequentist settings as the sampling distribution of test statistics, especially in maximum likelihood estimation settings.
We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.
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)
See Also
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)