Particle horizon

Template:Probability distribution In probability theory and statistics, the F-distribution is a continuous probability distribution. It is also known as Snedecor's F distribution or the Fisher-Snedecor distribution (after R.A. Fisher and George W. Snedecor). The F-distribution arises frequently as the null distribution of a test statistic, most notably in the analysis of variance; see F-test.

Definition

If a random variable X has an F-distribution with parameters d₁ and d₂, we write X ~ F(d₁, d₂). Then the probability density function for X is given by

${\begin{aligned}f(x;d_{1},d_{2})&={\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\\&={\frac {1}{\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\left({\frac {d_{1}}{d_{2}}}\right)^{\frac {d_{1}}{2}}x^{{\frac {d_{1}}{2}}-1}\left(1+{\frac {d_{1}}{d_{2}}}\,x\right)^{-{\frac {d_{1}+d_{2}}{2}}}\end{aligned}}$

for real x ≥ 0. Here $\mathrm {B}$ is the beta function. In many applications, the parameters d₁ and d₂ are positive integers, but the distribution is well-defined for positive real values of these parameters.

The cumulative distribution function is

$F(x;d_{1},d_{2})=I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right),$

where I is the regularized incomplete beta function.

The expectation, variance, and other details about the F(d₁, d₂) are given in the sidebox; for d₂ > 8, the excess kurtosis is

$\gamma _{2}=12{\frac {d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}$.

The k-th moment of an F(d₁, d₂) distribution exists and is finite only when 2k < d₂ and it is equal to

$\mu _{X}(k)=\left({\frac {d_{2}}{d_{1}}}\right)^{k}{\frac {\Gamma \left({\tfrac {d_{1}}{2}}+k\right)}{\Gamma \left({\tfrac {d_{1}}{2}}\right)}}{\frac {\Gamma \left({\tfrac {d_{2}}{2}}-k\right)}{\Gamma \left({\tfrac {d_{2}}{2}}\right)}}$

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., ). The correct expression is

$\varphi _{d_{1},d_{2}}^{F}(s)={\frac {\Gamma ({\frac {d_{1}+d_{2}}{2}})}{\Gamma ({\tfrac {d_{2}}{2}})}}U\!\left({\frac {d_{1}}{2}},1-{\frac {d_{2}}{2}},-{\frac {d_{2}}{d_{1}}}\imath s\right)$

where U(a, b, z) is the confluent hypergeometric function of the second kind.

Characterization

A random variate of the F-distribution with parameters d₁ and d₂ arises as the ratio of two appropriately scaled chi-squared variates:

$X={\frac {U_{1}/d_{1}}{U_{2}/d_{2}}}$

where

• U₁ and U₂ have chi-squared distributions with d₁ and d₂ degrees of freedom respectively, and
• U₁ and U₂ are independent.

In instances where the F-distribution is used, for example in the analysis of variance, independence of U₁ and U₂ might be demonstrated by applying Cochran's theorem.

Equivalently, the random variable of the F-distribution may also be written

$X={\frac {s_{1}^{2}}{\sigma _{1}^{2}}}\;/\;{\frac {s_{2}^{2}}{\sigma _{2}^{2}}}$

where s₁² and s₂² are the sums of squares S₁² and S₂² from two normal processes with variances σ₁² and σ₂² divided by the corresponding number of χ² degrees of freedom, d₁ and d₂ respectively.(see discussion)

In a Frequentist context, a scaled F-distribution therefore gives the probability p(s₁²/s₂² | σ₁², σ₂²), with the F distribution itself, without any scaling, applying where σ₁² is being taken equal to σ₂². 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 X has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of σ₁² and σ₂². In this context, a scaled F-distribution thus gives the posterior probability p(σ₂²/σ₁²|s₁², s₂²), where now the observed sums s₁² and s₂² are what are taken as known.

Generalization

A generalization of the (central) F-distribution is the noncentral F-distribution.

Related distributions and properties

• If $X\sim \chi _{d_{1}}^{2}$ and $Y\sim \chi _{d_{2}}^{2}$ are independent, then ${\frac {X/d_{1}}{Y/d_{2}}}\sim \mathrm {F} (d_{1},d_{2})$
• If $X\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)$ (Beta distribution) then ${\frac {d_{2}X}{d_{1}(1-X)}}\sim \operatorname {F} (d_{1},d_{2})$
• Equivalently, if X ~ F(d₁, d₂), then ${\frac {d_{1}X/d_{2}}{1+d_{1}X/d_{2}}}\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)$.
• If X ~ F(d₁, d₂) then $Y=\lim _{d_{2}\to \infty }d_{1}X$ has the chi-squared distribution $\chi _{d_{1}}^{2}$
• F(d₁, d₂) is equivalent to the scaled Hotelling's T-squared distribution ${\frac {d_{2}}{d_{1}(d_{1}+d_{2}-1)}}\operatorname {T} ^{2}(d_{1},d_{1}+d_{2}-1)$.
• If X ~ F(d₁, d₂) then X⁻¹ ~ F(d₂, d₁).
• If X ~ t(n) then

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

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

• F-distribution is a special case of type 6 Pearson distribution

• If X and Y are independent, with X, Y ~ Laplace(μ, b) then

${\tfrac {|X-\mu |}{|Y-\mu |}}\sim \operatorname {F} (2,2)$

• If X ~ F(n, m) then ${\tfrac {\log {X}}{2}}\sim \operatorname {FisherZ} (n,m)$ (Fisher's z-distribution)
• The noncentral F-distribution simplifies to the F-distribution if λ = 0.
• The doubly noncentral F-distribution simplifies to the F-distribution if $\lambda _{1}=\lambda _{2}=0$

• If $\operatorname {Q} _{X}(p)$ is the quantile p for X ~ F(d₁, d₂) and $\operatorname {Q} _{Y}(1-p)$ is the quantile 1−p for Y ~ F(d₂, d₁), then

$\operatorname {Q} _{X}(p)={\frac {1}{\operatorname {Q} _{Y}(1-p)}}$.

See also

• Chi-squared distribution
• Chow test
• Gamma distribution
• Hotelling's T-squared distribution
• Student's t-distribution
• Wilks' lambda distribution
• Wishart distribution

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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

55 yrs old Metal Polisher Records from Gypsumville, has interests which include owning an antique car, summoners war hack and spelunkering. Gets immense motivation from life by going to places such as Villa Adriana (Tivoli).

my web site - summoners war hack no survey ios Template:Common univariate probability distributions