ChiSquare {distributions3} | R Documentation |

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. |

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 `ChiSquare`

object.

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()`

,
`Erlang()`

,
`Exponential()`

,
`FisherF()`

,
`Frechet()`

,
`GEV()`

,
`GP()`

,
`Gamma()`

,
`Gumbel()`

,
`LogNormal()`

,
`Logistic()`

,
`Normal()`

,
`RevWeibull()`

,
`StudentsT()`

,
`Tukey()`

,
`Uniform()`

,
`Weibull()`

```
set.seed(27)
X <- ChiSquare(5)
X
mean(X)
variance(X)
skewness(X)
kurtosis(X)
random(X, 10)
pdf(X, 2)
log_pdf(X, 2)
cdf(X, 4)
quantile(X, 0.7)
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
```

[Package *distributions3* version 0.2.1 Index]