Description Usage Arguments Details Value Transformations See Also Examples

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

1 |

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

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 *χ^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) = 1 / (2 π σ^2) exp(-(x - μ)^2 / (2 σ^2))
*

**Cumulative distribution function (c.d.f)**:

The cumulative distribution function has the form

*
F(t) = integral_{-∞}^t 1 / (2 π σ^2) exp(-(x - μ)^2 / (2 σ^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 *Φ(t)* also stands
for the c.d.f. of a standard normal evaluated at *t*. Z-tables
list the value of *Φ(t)* for various *t*.

**Moment generating function (m.g.f)**:

*
E(e^(tX)) = e^(μ t + σ^2 t^2 / 2)
*

A `ChiSquare`

object.

A squared standard `Normal()`

distribution is equivalent to a
*χ^2_1* distribution with one degree of freedom. The
*χ^2* distribution is a special case of the `Gamma()`

distribution with shape (TODO: check this) parameter equal
to a half. Sums of *χ^2* distributions
are also distributed as *χ^2* distributions, where the
degrees of freedom of the contributing distributions get summed.
The ratio of two *χ^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`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.