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