F-Distribution

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A F-Distribution is a continuous probability distribution with parameters [math]d_1[/math] and [math]d_2[/math] such that [math] f(x; d_1,d_2) = [/math] [math] \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) }[/math] [math] = \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} }[/math]



References

2016

  • (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/F-distribution Retrieved:2016-8-7.
    • The F-distribution, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is, in probability theory and statistics, a continuous probability distribution.

      The F-distribution arises frequently as the null distribution of a test statistic, most notably in the analysis of variance; see F-test.

      • parameters =d1, d2 > 0 deg. of freedom|

        support = x ∈ [0, +∞)|

        pdf = [math] \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)}\! [/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 d2 > 2|

        median =|

        mode = [math] \frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2} [/math]
        for d1 > 2|

        variance = [math] \frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\! [/math]
        for d2 > 4|

        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 d2 > 6


  • (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/F-distribution#Definition Retrieved:2016-8-7.
    • 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 (pdf) for X is given by : [math] \begin{align} 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{align} [/math] for real x ≥ 0. Here [math] \mathrm{B} [/math] 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 : [math] 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 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 : [math] \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 k-th moment of an F(d1, d2) distribution exists and is finite only when 2k < d2 and it is equal to [1] : [math] \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.,[2]). The correct expression [3] is : [math] \varphi^F_{d_1, d_2}(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) [/math] where U(a, b, z) is the confluent hypergeometric function of the second kind.

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  2. Cite error: Invalid <ref> tag; no text was provided for refs named abramowitz
  3. Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261–264

2008