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

ChiSquare(df)

df | Degrees of freedom. Must be positive. |
---|

A `ChiSquare`

object.

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)**:

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

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.

Other continuous distributions: `Beta`

,
`Cauchy`

, `Exponential`

,
`FisherF`

, `Gamma`

,
`LogNormal`

, `Logistic`

,
`Normal`

, `StudentsT`

,
`Tukey`

, `Uniform`

,
`Weibull`

#> Chi Square distribution (df = 5)#> [1] 11.2129049 7.8935724 2.1298341 5.2084236 5.4563211 3.6636712 #> [7] 10.9823299 0.7858347 4.8748588 1.7938110#> [1] 0.1383692#> [1] -1.97783#> [1] 0.450584#> [1] 6.06443#> [1] 0.7#> [1] 7