Chi-square distributions show up often in frequentist settings as the sampling distribution of test statistics, especially in maximum likelihood estimation settings.
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.
Examples
set.seed(27)
X <- ChiSquare(5)
X
#> [1] "ChiSquare(df = 5)"
mean(X)
#> [1] 5
variance(X)
#> [1] 10
skewness(X)
#> [1] 1.264911
kurtosis(X)
#> [1] 2.4
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