F-Distribution

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]

• AKA: Snedecor's F-distribution, Fisher–Snedecor distribution.
• Context:
  • It can be referenced by an F-Statistic.
  • …
• Counter-Example(s):
  • a Beta Distribution.
  • a Beta Prime Distribution.
  • a z Distribution.
  • a t-Test Distribution.
• See: F-test, F-test of Equality of Variances, Analysis of variance, Statistical Test, Bartlett's Test, Levene's Test, Brown–Forsythe Test, Population Genetics, One-way ANOVA, Null Distribution, Confluent Hypergeometric Function, Beta Function, Excess Kurtosis.

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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 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.

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
  • QUOTE: F-Distribution: If [math]\displaystyle{ Y_1 }[/math] and [math]\displaystyle{ Y_2 }[/math] are independent chi-squared random variables with [math]\displaystyle{ v_1 }[/math] and [math]\displaystyle{ v_2 }[/math] degrees of freedom, respectively, then the ratio [math]\displaystyle{ X }[/math] given by :[math]\displaystyle{ X = \frac{Y_1}{v_1}\bigg / \frac{Y_2}{v_2} }[/math] is said to have an [math]\displaystyle{ F }[/math]-distribution with [math]\displaystyle{ v_1 }[/math], and [math]\displaystyle{ v_2 }[/math] degrees of freedom. This may be written as the [math]\displaystyle{ F_{v_1, v_2} }[/math]-distribution. Evidently 1/X will have a [math]\displaystyle{ F_{v_2, v_1} }[/math]-distribution.

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