distributional (version 0.2.1)

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

Description

stable

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 ${\chi }^2$ 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

$$F(t)={\int }_{-\mathrm{\infty }}^t\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-(x-\mu {)}^2/2{\sigma }^2}dx$$

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 $\mathrm{\Phi }(t)$ also stands for the c.d.f. of a standard normal evaluated at $t$. Z-tables list the value of $\mathrm{\Phi }(t)$ 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

# NOT RUN {
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)

# }