F-Distribution

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A F-Distribution is a continuous probability distribution with parameters [math]\displaystyle{ d_1 }[/math] and [math]\displaystyle{ d_2 }[/math] such that [math]\displaystyle{ f(x; d_1,d_2) = }[/math] [math]\displaystyle{ \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]\displaystyle{ = \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]\displaystyle{ \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]\displaystyle{ I_{\frac{d_1 x}{d_1 x + d_2}} \left(\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) }[/math] |

        mean = [math]\displaystyle{ \frac{d_2}{d_2-2}\! }[/math]
        for d2 > 2|

        median =|

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

        variance = [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \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  : [math]\displaystyle{ \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., ). The correct expression [1] is : [math]\displaystyle{ \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.

  1. Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261–264

2008