Chi-square distributions show up often in frequentist settings as the sampling distribution of test statistics, especially in maximum likelihood estimation settings.

ChiSquare(df)

Arguments

df

Degrees of freedom. Must be positive.

Value

A ChiSquare object.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.

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):

$$ E(e^{tX}) = e^{\mu t + \sigma^2 t^2 / 2} $$

Transformations

A squared standard Normal() distribution is equivalent to a \(\chi^2_1\) distribution with one degree of freedom. The \(\chi^2\) distribution is a special case of the Gamma() distribution with shape (TODO: check this) parameter equal to a half. Sums of \(\chi^2\) distributions are also distributed as \(\chi^2\) distributions, where the degrees of freedom of the contributing distributions get summed. The ratio of two \(\chi^2\) distributions is a FisherF() distribution. The ratio of a Normal() and the square root of a scaled ChiSquare() is a StudentsT() distribution.

See also

Examples

set.seed(27) X <- ChiSquare(5) X
#> Chi Square distribution (df = 5)
random(X, 10)
#> [1] 11.2129049 7.8935724 2.1298341 5.2084236 5.4563211 3.6636712 #> [7] 10.9823299 0.7858347 4.8748588 1.7938110
pdf(X, 2)
#> [1] 0.1383692
log_pdf(X, 2)
#> [1] -1.97783
cdf(X, 4)
#> [1] 0.450584
quantile(X, 0.7)
#> [1] 6.06443
cdf(X, quantile(X, 0.7))
#> [1] 0.7
quantile(X, cdf(X, 7))
#> [1] 7