Stable lifecycle

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

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

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

Examples

dist <- dist_chisq(df = c(1,2,3,4,6,9)) dist
#> <distribution[6]> #> [1] \u1d6a²(1) \u1d6a²(2) \u1d6a²(3) \u1d6a²(4) \u1d6a²(6) \u1d6a²(9)
mean(dist)
#> [1] 1 2 3 4 6 9
variance(dist)
#> [1] 2 4 6 8 12 18
skewness(dist)
#> [1] 2.828427 2.000000 1.632993 1.414214 1.154701 0.942809
kurtosis(dist)
#> [1] 12.000000 6.000000 4.000000 3.000000 2.000000 1.333333
generate(dist, 10)
#> [[1]] #> [1] 0.72601085 0.93071502 0.02579683 0.08327627 0.42007532 1.86026126 #> [7] 0.77809157 0.11670245 2.61405677 0.02478708 #> #> [[2]] #> [1] 0.6910968 4.3422720 1.6989898 1.1801349 5.1658135 5.6831134 0.6983211 #> [8] 1.4066345 4.9075193 0.4674858 #> #> [[3]] #> [1] 3.169329 1.799478 2.351222 2.506705 2.500567 1.515333 1.164510 #> [8] 4.038398 15.613134 2.493755 #> #> [[4]] #> [1] 6.1652771 6.4518952 6.3843386 2.1063540 8.1999058 6.5853996 8.9435799 #> [8] 2.3703962 2.2069658 0.9125746 #> #> [[5]] #> [1] 4.982996 16.885079 2.017798 4.772882 1.789387 5.017964 3.160282 #> [8] 6.399912 1.462012 7.077559 #> #> [[6]] #> [1] 8.236767 3.672691 8.504930 4.504704 8.915198 5.628488 11.086178 #> [8] 3.053023 6.015485 11.267140 #>
density(dist, 2)
#> [1] 0.10377687 0.18393972 0.20755375 0.18393972 0.09196986 0.01581362
density(dist, 2, log = TRUE)
#> [1] -2.265512 -1.693147 -1.572365 -1.693147 -2.386294 -4.146884
cdf(dist, 4)
#> [1] 0.95449974 0.86466472 0.73853587 0.59399415 0.32332358 0.08858747
quantile(dist, 0.7)
#> [1] 1.074194 2.407946 3.664871 4.878433 7.231135 10.656372