class

Chi2

extendsGamma

Chi2(df: Tensor | float, validate_args: bool | None = None)

source

Chi-squared distribution with $k$ degrees of freedom.

Distribution of the sum of squares of $k$ independent standard Normal random variables. Foundational in classical statistics: arises in the likelihood-ratio test (Wilks' theorem), Pearson's goodness-of-fit, confidence intervals for variance, and many F-test /t-test derivations.

Equivalent to a Gamma with shape $k/2$ and rate $1/2$ : $\chi^2(k) = \mathrm{Gamma}(k/2,\, 1/2)$ .

Parameters

dfTensor or float

Degrees of freedom

k > 0

. Need not be an integer.

validate_argsbool= None

If True, validate parameter constraints at construction time.

Notes

Probability density on $x > 0$ :

p(x; k) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{k/2 - 1} e^{-x/2}

Moments:

\mathbb{E}[X] = k, \qquad \mathrm{Var}[X] = 2k, \qquad \mathrm{Mode} = \max(k - 2, 0)

Special cases:

$k = 1$ → squared standard Normal.
$k = 2$ → $\mathrm{Exponential}(1/2)$ .
Sum: $\chi^2(k_1) + \chi^2(k_2) \sim \chi^2(k_1 + k_2)$ .

By the central limit theorem, $(\chi^2(k) - k)/\sqrt{2k} \to \mathcal{N}(0, 1)$ as $k \to \infty$ .

Examples

>>> import lucid
>>> from lucid.distributions import Chi2
>>> d = Chi2(df=4.0)
>>> d.mean
Tensor(4.0)
>>> d.sample((4,))
Tensor([...])

Methods (1)

dunder

init

→None

__init__(df: Tensor | float, validate_args: bool | None = None)

source

Construct a Chi-squared distribution.

Parameters

dfTensor | float

Degrees of freedom

k > 0

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Notes

$\chi^2(k)$ is a special case of the Gamma distribution:

\chi^2(k) = \text{Gamma}\!\left(\frac{k}{2},\; \frac{1}{2}\right)

It arises as the distribution of the sum of squares of $k$ independent standard Normal variables, and is fundamental in hypothesis testing (e.g., goodness-of-fit tests, likelihood-ratio tests).

Examples

>>> from lucid.distributions import Chi2
>>> d = Chi2(df=4.0)
>>> d.mean  # k = 4
Tensor(4.0)