F-distribution

{(d_1 x+d_2)^{d_1+d_2{x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}</math>

| cdf = <math>I_{\frac{d_1 x}{d_1 x + d_2 \left(\tfrac{d_1}{2}, \tfrac{d_2}{2} \right)</math>

| mean = <math>\frac{d_2}{d_2-2}</math> for

| median =

| mode = <math>\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}</math><br /> for

| variance = <math>\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}</math> for

| skewness = <math>\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4){(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)</math> for

| kurtosis = see text

| entropy = <math>\begin{align}

& \ln \Gamma{\left(\tfrac{d_1}{2} \right)}

+ \ln \Gamma{\left(\tfrac{d_2}{2} \right)}

- \ln \Gamma{\left(\tfrac{d_1+d_2}{2} \right)} \\

&+ \left(1-\tfrac{d_1}{2} \right) \psi{\left(1+\tfrac{d_1}{2} \right)} \\

&- \left(1+\tfrac{d_2}{2} \right) \psi{\left(1+\tfrac{d_2}{2} \right)} \\

&+ \left(\tfrac{d_1 + d_2}{2} \right) \psi{\left(\tfrac{d_1 + d_2}{2} \right)}

+ \ln \frac{d_2}{d_1}

\end{align}</math>

| mgf = does not exist, raw moments defined in text and in

Definitions

The F-distribution with and degrees of freedom is the distribution of

where <math display=inline>U_1</math> and <math display=inline>U_2</math> are independent random variables with chi-square distributions with respective degrees of freedom and .

It can be shown to follow that the probability density function () for is given by

<math display="block">\begin{align}

f(x; d_1,d_2) &= \frac{\sqrt{\frac{(d_1x)^{d_1}\,\,d_2^{d_2 {(d_1x+d_2)^{d_1+d_2 {x\operatorname{B}\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \\[5pt]

&=\frac{\left(\frac{d_1}{d_2}\right)^{\frac{d_1}{2 {x\vphantom{\left({d_1\over d_2}\right)^{\frac{d_1}{2} - 1} \left(1+\frac{d_1}{d_2} \, x \right)^{-\frac{d_1+d_2}{2}{\operatorname{B}\left(\frac{d_1}{2},\frac{d_2}{2}\right)}

\end{align}</math>

for real . Here <math>\mathrm{B}</math> is the beta function. In many applications, the parameters and are positive integers, but the distribution is well-defined for positive real values of these parameters.

The cumulative distribution function is

<math display="block">F(x; d_1,d_2)=I_\frac{d_1 x}{(d_1 x + d_2)}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) ,</math>

where is the regularized incomplete beta function.

Properties

The expectation, variance, and other details about the F-distribution are given in the sidebox; for , the excess kurtosis is

<math display="block">\gamma_2 = 12\frac{d_1(5d_2-22)(d_1+d_2-2)+(d_2-4)(d_2-2)^2}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}.</math>

The -th moment of an distribution exists and is finite only when and it is equal to

<math display="block">\mu _X(k) =\left( \frac{d_2}{d_1}\right)^k \frac{\Gamma \left(\tfrac{d_1}{2}+k\right) }{\Gamma \left(\tfrac{d_1}{2}\right)} \frac{\Gamma \left(\tfrac{d_2}{2}-k\right) }{\Gamma \left( \tfrac{d_2}{2}\right) }.</math>

The F-distribution is a particular parametrization of the beta prime distribution, which is also called the beta distribution of the second kind.

The characteristic function is listed incorrectly in many standard references (e.g., is

<math display="block">\varphi^F_{d_1, d_2}(s) = \frac{\Gamma{\left(\frac{d_1+d_2}{2}\right){\Gamma{\left(\tfrac{d_2}{2}\right) U \! \left(\frac{d_1}{2},1-\frac{d_2}{2},-\frac{d_2}{d_1} \imath s \right)</math>

where is the confluent hypergeometric function of the second kind.

Relation to the chi-squared distribution

In instances where the F-distribution is used, for example in the analysis of variance, independence of and (defined above) might be demonstrated by applying Cochran's theorem.

Equivalently, since the chi-squared distribution is the sum of squares of independent standard normal random variables, the random variable of the F-distribution may also be written

<math display="block">X = \frac{s_1^2}{\sigma_1^2} \div \frac{s_2^2}{\sigma_2^2},</math>

where <math display="inline">s_1^2 = \frac{S_1^2}{d_1}</math> and , <math>S_1^2</math> is the sum of squares of <math>d_1</math> random variables from normal distribution <math>N(0,\sigma_1^2)</math> and <math>S_2^2</math> is the sum of squares of <math>d_2</math> random variables from normal distribution .

In a frequentist context, a scaled F-distribution therefore gives the probability , with the F-distribution itself, without any scaling, applying where <math>\sigma_1^2</math> is being taken equal to . This is the context in which the F-distribution most generally appears in F-tests: where the null hypothesis is that two independent normal variances are equal, and the observed sums of some appropriately selected squares are then examined to see whether their ratio is significantly incompatible with this null hypothesis.

The quantity <math>X</math> has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of <math>\sigma_1^2</math> and . In this context, a scaled F-distribution thus gives the posterior probability , where the observed sums <math>s^2_1</math> and <math>s^2_2</math> are now taken as known.

In general

If <math>X \sim \chi^2_{d_1}</math> and <math>Y \sim \chi^2_{d_2}</math> (Chi squared distribution) are independent, then .
If <math>X_k \sim \Gamma(\alpha_k,\beta_k)\,</math> (Gamma distribution) are independent, then .
If <math display="inline">X \sim \operatorname{Beta}\left({d_1\over2},{d_2\over2}\right)</math> (Beta distribution) then .
Equivalently, if , then .
If , then <math display="inline">\frac{d_1}{d_2}X</math> has a beta prime distribution: .
If <math>X \sim F(d_1, d_2)</math> then <math display="inline">Y = \lim_{d_2 \to \infty} d_1 X</math> has the chi-squared distribution .
<math>F(d_1, d_2)</math> is equivalent to the scaled Hotelling's T-squared distribution .
If <math>X \sim F(d_1, d_2)</math> then .
If Student's t-distribution then: <math display="block">\begin{align}

X^{2} &\sim \operatorname{F}(1, n), \\

X^{-2} &\sim \operatorname{F}(n, 1).

\end{align}</math>

F-distribution is a special case of type 6 Pearson distribution.
If <math>X</math> and <math>Y</math> are independent, with (Laplace distribution), then <math display="block"> \frac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2). </math>
If <math>X\sim F(n,m)</math> then <math>\tfrac{\log{X{2} \sim \operatorname{FisherZ}(n,m)</math> (Fisher's z-distribution).
The noncentral F-distribution simplifies to the F-distribution if .
The doubly noncentral F-distribution simplifies to the F-distribution if
If <math>\operatorname{Q}_X(p)</math> is the quantile for <math>X\sim F(d_1,d_2)</math> and <math>\operatorname{Q}_Y(1-p)</math> is the quantile <math>1-p</math> for , then <math display="block">\operatorname{Q}_X(p)=\frac{1}{\operatorname{Q}_Y(1-p)}.</math>
F-distribution is an instance of ratio distributions.
W-distribution is a unique parametrization of F-distribution.

References

External links

Table of critical values of the F-distribution
Earliest Uses of Some of the Words of Mathematics: entry on F-distribution contains a brief history
Free calculator for F-testing

Definitions

Properties

Related distributions

Relation to the chi-squared distribution

In general

See also

References

External links