class

FisherSnedecor

extendsDistribution

FisherSnedecor(df1: Tensor | float, df2: Tensor | float, validate_args: bool | None = None)

source edit

Fisher-Snedecor F-distribution — ratio of two scaled Chi-squared variates.

FisherSnedecor(df1=d1, df2=d2) is the distribution of the statistic

F = \frac{X / d_1}{Y / d_2}

where $X \sim \chi^2(d_1)$ and $Y \sim \chi^2(d_2)$ are independent. The F-distribution is foundational in classical statistics: it arises in ANOVA F-tests and in the likelihood-ratio test for comparing nested linear models.

Parameters

df1Tensor | float

Numerator degrees of freedom

d_1 > 0

df2Tensor | float

Denominator degrees of freedom

d_2 > 0

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

df1Tensor

Numerator degrees of freedom

d_1

df2Tensor

Denominator degrees of freedom

d_2

Notes

PDF:

p(x; d_1, d_2) = \frac{1}{x \, B(d_1/2,\, d_2/2)} \left(\frac{d_1 x}{d_1 x + d_2}\right)^{d_1/2} \left(\frac{d_2}{d_1 x + d_2}\right)^{d_2/2}, \quad x > 0

Log-PDF (stable form using lgamma):

\log p(x) = \frac{d_1}{2}\log(d_1 x) + \frac{d_2}{2}\log d_2 - \frac{d_1+d_2}{2}\log(d_1 x + d_2) - \log x - \log B(d_1/2, d_2/2)

Moments (when defined):

Mean ( $d_2 > 2$ ): $E[X] = d_2 / (d_2 - 2)$
Variance ( $d_2 > 4$ ): $\operatorname{Var}[X] = 2 d_2^2 (d_1 + d_2 - 2) / [d_1 (d_2 - 2)^2 (d_2 - 4)]$

Sampling draws one $\chi^2(d_1)$ and one $\chi^2(d_2)$ sample and forms the ratio. Both $\chi^2$ samples are drawn via the Gamma sampler.

Examples

>>> import lucid
>>> from lucid.distributions import FisherSnedecor
>>> dist = FisherSnedecor(df1=5.0, df2=10.0)
>>> samples = dist.sample((100,))
>>> dist.mean  # d2/(d2-2) = 10/8 = 1.25

Used by 1

lucid.distributions

Constructors

dunder

init

→None

__init__(df1: Tensor | float, df2: Tensor | float, validate_args: bool | None = None)

source edit

Initialise a Fisher-Snedecor F-distribution.

Parameters

df1Tensor | float

Numerator degrees of freedom

d_1 > 0

df2Tensor | float

Denominator degrees of freedom

d_2 > 0

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Mean of the F-distribution: $E[X] = d_2 / (d_2 - 2)$ .

Defined only for $d_2 > 2$ ; returns inf or nan otherwise.

Returns

Tensor

Mean values of shape batch_shape.

prop

variance

→Tensor

variance: Tensor

source edit

Variance of the F-distribution.

$\operatorname{Var}[X] = 2 d_2^2 (d_1 + d_2 - 2) / [d_1 (d_2 - 2)^2 (d_2 - 4)]$

Defined only for $d_2 > 4$ .

Returns

Tensor

Variance values of shape batch_shape.

Instance methods

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the F-distribution.

\log p(x) = \frac{d_1}{2}\log(d_1 x) + \frac{d_2}{2}\log d_2 - \frac{d_1+d_2}{2}\log(d_1 x + d_2) - \log x - \log B(d_1/2, d_2/2)

Parameters

valueTensor

Positive points

x > 0

at which to evaluate the density.

Returns

Tensor

Log-density values of the same shape as value.

sample

→Tensor

sample(sample_shape: tuple[int, ...] = ())

source edit

Draw samples from the F-distribution.

Forms the ratio $F = (X/d_1) / (Y/d_2)$ where $X \sim \chi^2(d_1)$ and $Y \sim \chi^2(d_2)$ are independent samples drawn via the Gamma sampler.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Positive samples of shape (*sample_shape, *batch_shape).

class

FisherSnedecor

extendsDistribution

FisherSnedecor(df1: Tensor | float, df2: Tensor | float, validate_args: bool | None = None)

source edit

Fisher-Snedecor F-distribution — ratio of two scaled Chi-squared variates.

FisherSnedecor(df1=d1, df2=d2) is the distribution of the statistic

F = \frac{X / d_1}{Y / d_2}

Parameters

df1Tensor | float

Numerator degrees of freedom

d_1 > 0

df2Tensor | float

Denominator degrees of freedom

d_2 > 0

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

df1Tensor

Numerator degrees of freedom

d_1

df2Tensor

Denominator degrees of freedom

d_2

Notes

PDF:

p(x; d_1, d_2) = \frac{1}{x \, B(d_1/2,\, d_2/2)} \left(\frac{d_1 x}{d_1 x + d_2}\right)^{d_1/2} \left(\frac{d_2}{d_1 x + d_2}\right)^{d_2/2}, \quad x > 0

Log-PDF (stable form using lgamma):

\log p(x) = \frac{d_1}{2}\log(d_1 x) + \frac{d_2}{2}\log d_2 - \frac{d_1+d_2}{2}\log(d_1 x + d_2) - \log x - \log B(d_1/2, d_2/2)

Moments (when defined):

Mean ( $d_2 > 2$ ): $E[X] = d_2 / (d_2 - 2)$
Variance ( $d_2 > 4$ ): $\operatorname{Var}[X] = 2 d_2^2 (d_1 + d_2 - 2) / [d_1 (d_2 - 2)^2 (d_2 - 4)]$

Sampling draws one $\chi^2(d_1)$ and one $\chi^2(d_2)$ sample and forms the ratio. Both $\chi^2$ samples are drawn via the Gamma sampler.

Examples

>>> import lucid
>>> from lucid.distributions import FisherSnedecor
>>> dist = FisherSnedecor(df1=5.0, df2=10.0)
>>> samples = dist.sample((100,))
>>> dist.mean  # d2/(d2-2) = 10/8 = 1.25

Used by 1

lucid.distributions

Constructors

dunder

init

→None

__init__(df1: Tensor | float, df2: Tensor | float, validate_args: bool | None = None)

source edit

Initialise a Fisher-Snedecor F-distribution.

Parameters

df1Tensor | float

Numerator degrees of freedom

d_1 > 0

df2Tensor | float

Denominator degrees of freedom

d_2 > 0

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Mean of the F-distribution: $E[X] = d_2 / (d_2 - 2)$ .

Defined only for $d_2 > 2$ ; returns inf or nan otherwise.

Returns

Tensor

Mean values of shape batch_shape.

prop

variance

→Tensor

variance: Tensor

source edit

Variance of the F-distribution.

$\operatorname{Var}[X] = 2 d_2^2 (d_1 + d_2 - 2) / [d_1 (d_2 - 2)^2 (d_2 - 4)]$

Defined only for $d_2 > 4$ .

Returns

Tensor

Variance values of shape batch_shape.

Instance methods

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the F-distribution.

\log p(x) = \frac{d_1}{2}\log(d_1 x) + \frac{d_2}{2}\log d_2 - \frac{d_1+d_2}{2}\log(d_1 x + d_2) - \log x - \log B(d_1/2, d_2/2)

Parameters

valueTensor

Positive points

x > 0

at which to evaluate the density.

Returns

Tensor

Log-density values of the same shape as value.

sample

→Tensor

sample(sample_shape: tuple[int, ...] = ())

source edit

Draw samples from the F-distribution.

Forms the ratio $F = (X/d_1) / (Y/d_2)$ where $X \sim \chi^2(d_1)$ and $Y \sim \chi^2(d_2)$ are independent samples drawn via the Gamma sampler.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Positive samples of shape (*sample_shape, *batch_shape).