F-Distribution

Table of contents

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Density Function

The density function of the F distribution:

\[f(x; d_1, d_2) = \frac{1}{\mathcal{B} \left( \frac{d_1}{2}, \frac{d_2}{2} \right)} \left( \frac{d_1}{d_2} \right)^{\frac{d_1}{2}} x^{d_1/2 - 1} \left(1 + \frac{d_1}{d_2} x \right)^{- \frac{d_1+d_2}{2}} \times \mathbf{1} [x \geq 0]\]

where \(\mathcal{B}(a,b)\) denotes the Beta function.

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.df(x: float, df1: float = 1.0, df2: float = 1.0, log: bool = False) → float

Density function of the F-distribution distribution.

Example

>>> pystats.df(1.5, 10.0, 12.0)
0.3426270538333347

Parameters

• x (float) – A real-valued input.

• df1 (float) – A degrees of freedom parameter, a real-valued input.

• df2 (float) – A degrees of freedom parameter, a real-valued input.

• log (bool) – Return the log-density or the true form.

Returns

The density function evaluated at x.

List Input

pystats.df(x: List[float], df1: float = 1.0, df2: float = 1.0, log: bool = False) → List[float]

Density function of the F-distribution distribution.

Example

>>> pystats.df([0.3, 0.5, 0.9], 10.0, 12.0)
[0.3523215359999982, 0.6861197229627093, 0.7047187344898975]

Parameters

• x (List[float]) – A standard list input.

• df1 (float) – A degrees of freedom parameter, a real-valued input.

• df2 (float) – A degrees of freedom parameter, a real-valued input.

• log (bool) – Return the log-density or the true form.

Returns

A list of density values corresponding to the elements of x.

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Cumulative Distribution Function

The cumulative distribution function (CDF) of the F distribution:

\[F(x; d_1, d_2) = \int_0^x f(z; d_1, d_2) dz = I_{\frac{d_1 x}{d_2 + d_1 x}} (d_1 / 2, d_2 / 2)\]

where \(I_x (a,b)\) denotes the regularized incomplete Beta function.

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.pf(p: float, df1: float = 1.0, df2: float = 1.0, log: bool = False) → float

Distribution function of the F-distribution distribution.

Example

>>> pystats.pf(1.5, 10.0, 12.0)
0.7501297253279772

Parameters

• p (float) – A real-valued input.

• df1 (float) – A degrees of freedom parameter, a real-valued input.

• df2 (float) – A degrees of freedom parameter, a real-valued input.

• log (bool) – Return the log-density or the true form.

Returns

The cumulative distribution function evaluated at p.

List Input

pystats.pf(p: List[float], df1: float = 1.0, df2: float = 1.0, log: bool = False) → List[float]

Distribution function of the F-distribution distribution.

Example

>>> pystats.pf([0.3, 0.5, 0.9], 10.0, 12.0)
[0.03279349759999985, 0.14036282082102308, 0.43990811032084115]

Parameters

• p (List[float]) – A standard list input.

• df1 (float) – A degrees of freedom parameter, a real-valued input.

• df2 (float) – A degrees of freedom parameter, a real-valued input.

• log (bool) – Return the log-density or the true form.

Returns

A list of CDF values corresponding to the elements of p.

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Quantile Function

The quantile function of the F distribution:

\[q(p; a,b) = \inf \left\{ x : p \leq I_{\frac{d_1 x}{d_2 + d_1 x}} (d_1 / 2, d_2 / 2) \right\}\]

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.qf(q: float, df1: float = 1.0, df2: float = 1.0) → float

Quantile function of the F-distribution distribution.

Example

>>> pystats.qf(0.5, 10.0, 12.0)
0.9885595669294069

Parameters

• q (float) – A real-valued input.

• df1 (float) – A degrees of freedom parameter, a real-valued input.

• df2 (float) – A degrees of freedom parameter, a real-valued input.

Returns

The quantile function evaluated at q.

List Input

pystats.qf(q: List[float], df1: float = 1.0, df2: float = 1.0) → List[float]

Quantile function of the F-distribution distribution.

Example

>>> pystats.qf([0.3, 0.5, 0.9], 10.0, 12.0)
[0.7125992144145168, 0.9885595669294069, 2.1877640788750874]

Parameters

• q (List[float]) – A standard list input.

• df1 (float) – A degrees of freedom parameter, a real-valued input.

• df2 (float) – A degrees of freedom parameter, a real-valued input.

Returns

A list of quantiles values corresponding to the elements of q.

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Random Sampling

Random sampling for the Beta distribution is achieved by simulating two independent \(\chi^2\)-distributed random variables, \(X \sim \chi^2(d_1), Y \sim \chi^2(d_2)\), then returning:

\[Z = \frac{d_1}{d_2} \frac{X}{Y}\]

Scalar Output

pystats.rf(df1: float = 1.0, df2: float = 1.0) → float

Random sampling function for the F-distribution distribution.

Example

>>> pystats.rf(10.0, 12.0)
2.646874442851056

Parameters

• df1 (float) – A degrees of freedom parameter, a real-valued input.

• df2 (float) – A degrees of freedom parameter, a real-valued input.

Returns

A pseudo-random draw from the F-distribution distribution.

List Output

pystats.rf(n: int, df1: float = 1.0, df2: float = 1.0) → List[float]

Random sampling function for the F-distribution distribution.

Example

>>> pystats.rf(3, 10.0, 12.0)
[0.4932413364221738, 0.2827026830899671, 2.750316525821291]

Parameters

• n (int) – The number of output values.

• df1 (float) – A degrees of freedom parameter, a real-valued input.

• df2 (float) – A degrees of freedom parameter, a real-valued input.

Returns

A list of pseudo-random draws from the F-distribution distribution.