﻿ F-distribution

# F-distribution

The F distribution or the Fisher distribution, and Fisher - Snedecor distribution (after Ronald Aylmer Fisher and George W. Snedecor ), is a continuous probability distribution. An F- distributed random variable is calculated as the quotient of two each divided by the corresponding number of degrees of freedom chi-square distributed random variables. The F-distribution has two independent degrees of freedom as a parameter, and thus forms a two- parameter distribution family.

The F distribution is often used in a test (F- test) to determine whether the difference between two sample variances based on a statistical fluctuation or whether it refers to different populations. Also in the context of analysis of variance is tested with an F statistic significant differences between populations ( groups).

• 3.1 Relationship to the beta distribution
• 3.2 Relationship to the chi -square distribution
• 3.3 Relationship to the non-central F distribution 3.3.1 Density of the non-central F distribution

## Definition

A continuous random variable satisfies the F- distribution with m degrees of freedom in the numerator and n degrees of freedom in the denominator, when the probability density

Possesses. This is referred to as the gamma function at the point.

Historically forms the definition below the origin of the F-distribution as the distribution of the size

Where and are independent, χ ² - distributed random variables with and degrees of freedom and degrees of freedom.

## Properties

### Expected value

The expected value is only defined and then reads

### Variance

The variance is only defined and then reads

### Distribution function

The values ​​of the distribution are usually determined numerically indicated in a table. A complete tabulation with respect to all degrees of freedom is ia not necessary, so that most of the distribution tables specify the quantiles with respect to selected degrees of freedom and probabilities. One does also the relationship advantage:

Where the quantile of the F distribution with means and degrees of freedom.

The F distribution can be expressed as a closed

The regularized incomplete beta function represents.

### Maximum

For taking on the site

The maximum.

### Entropy

The entropy of the F-distribution (expressed in nats ) is

Where ψ ( p) is the digamma - function referred.

## Relations with other distributions

The character means ' the ' is distributed as '.

### Relationship to the beta distribution

The random variable

Is beta distributed with parameters and applies:

Where and are independent chi-square distributed random variables with and degrees of freedom.

### Relationship with the chi -square distribution

From the independent and chi-square distributed random variables with degrees of freedom and can be

. construct This random variable is - distributed.

### Relation to non-central F distribution

For independent random variables and is

Distributed according to the non-central F- distribution with non- centrality parameter. It is a non-central Chi -square distribution with a non- centrality parameter and degrees of freedom. For there is the central F distribution.

#### Density of the non-central F distribution

The function is a special hypergeometric function, also known as the Kummer function and represents the above mentioned density of the central F- distribution.

Expected value and variance of the non-central F-distribution are given by

And

Both result from the formulas of the central F- distribution.

### Relation to the normal distribution

When the independent normal random variable parameters

Possess are the respective sample variances and independent, and we have:

And

Therefore, subject to the random variable

An F distribution with degrees of freedom in the numerator and the denominator degrees of freedom,

### Relationship to Student's t-distribution

If ( Student's t-distribution), then

The square of a t- distributed random variable with degrees of freedom follows an F distribution with and degrees of freedom.

## Derivation of the density

The probability density of the F distribution can be derived (see the derivation of the density of the Student's t-distribution) from the joint density of the two independent random variables, both of which are Chi -squared distributed.

The transformation

You get the joint density of and, where and.

The Jacobian of the transformation is:

The value is not important because it is multiplied by the calculation of the determinant to 0. The new density function is written like

Wanted is now the marginal distribution as an integral over the non- interest variable:

## Quantilfunktionen

The quantile of the F distribution is the solution of the equation, thus in principle be calculated via the inverse function. Specifically applies here

With the inverse of the regularized incomplete beta function. This value is in the F- distribution table registered under the coordinates, and.

For some values ​​, the Quantilsfunktionen can explicitly calculate. Dissolve the beta integral with which occur for a few indices invertible functions:

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