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F Distribution

Shape of Distribution​

Basic Properties​

  • Two parameters N1N_1 and N2N_2 are required (Positive integer)
  • Continuous distribution defined on semi-bounded range x≥0x \geq 0
  • This distribution is asymmetric.

Probability​

  • Probability density function

    f(x)=Γ(N1+N22)(N1N2)N12xN12−1Γ(N12)Γ(N22)(1+N1N2x)N1+N22f(x)=\frac{\Gamma\left(\frac{N_1+N_2}{2}\right)\left(\frac{N_1}{N_2}\right)^\frac{N_1}{2}x^{\frac{N_1}{2}-1}}{\Gamma\left(\frac{N_1}{2}\right)\Gamma\left(\frac{N_2}{2}\right)\left(1+\frac{N_1}{N_2}x\right)^{\frac{N_1+N_2}{2}}}

    , where Γ(⋅)\Gamma(\cdot) is gamma function.

  • Cumulative distribution function

    F(x)=Iγ(N12,N22)F(x)=I_\gamma\left(\frac{N_1}{2},\frac{N_2}{2}\right)

    , where γ=N1xN2+N1x\gamma=\frac{N_1x}{N_2+N_1x} and Ix(⋅,⋅)I_{x}(\cdot,\cdot) is regularized incomplete beta function.

  • How to compute these on Excel.

AB
1DataDescription
25Value for which you want the distribution
34Value of parameter N1
430Value of parameter N2
5FormulaDescription (Result)
6=NTFDIST(A2,A3,A4,TRUE)Cumulative distribution function for the terms above
7=NTFDIST(A2,A3,A4,FALSE)Probability density function for the terms above

Characteristics​

Mean -- Where is the "center" of the distribution? (Definition)​

  • Mean of the distribution is given as

    N2N2−2(N2>2)\frac{N_2}{N_2-2}\quad (N_2>2)

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter N2
3FormulaDescription (Result)
4=NTFMEAN(A2)Mean of the distribution for the terms above

Standard Deviation -- How wide does the distribution spread? (Definition)​

  • Variance of the distribution is given as

    2N22(N1+N2−2)N1(N2−2)2(N2−4)(N2>4)\frac{2N_2^2(N_1+N_2-2)}{N_1(N_2-2)^2(N_2-4)}\quad (N_2>4)

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel

AB
1DataDescription
24Value of parameter N1
330Value of parameter N2
4FormulaDescription (Result)
5=NTFSTDEV(A2,A3)Standard deviation of the distribution for the terms above

Skewness -- Which side is the distribution distorted into? (Definition)​

  • Skewness of the distribution is given as

    (2N1+N2−2)8(N2−4)N1(N1+N2−2)(N2−6)(N2>6)\frac{(2N_1+N_2-2)\sqrt{8(N_2-4)}}{\sqrt{N_1(N_1+N_2-2)}(N_2-6)}\quad (N_2>6)

  • How to compute this on Excel

AB
1DataDescription
24Value of parameter N1
330Value of parameter N2
4FormulaDescription (Result)
5=NTFSKEW(A2,A3)Skewness of the distribution for the terms above

Kurtosis -- Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)​

  • Kurtosis of the distribution is given as

    12[(N2−2)2(N2−4)+N1(N1+N2−2)(5N2−22)]N1(N2−6)(N2−8)(N1+N2−2)(N2>8)\frac{12[(N_2-2)^2(N_2-4)+N_1(N_1+N_2-2)(5N_2-22)]}{N_1(N_2-6)(N_2-8)(N_1+N_2-2)}\quad (N_2>8)

  • This distribution can be leptokurtic or platykurtic.

  • How to compute this on Excel

AB
1DataDescription
24Value of parameter N1
330Value of parameter N2
4FormulaDescription (Result)
5=NTFKURT(A2,A3)Kurtosis of the distribution for the terms above

Random Numbers​

  • How to generate random numbers on Excel.
AB
1DataDescription
24Value of parameter N1
330Value of parameter N2
4FormulaDescription (Result)
5=NTRANDF(100,A2,A3,0)100 F deviates based on Mersenne-Twister algorithm for which the parameters above

Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions​

  • If you already have parameters of the distribution
    • Generating random numbers based on Mersenne Twister algorithm: NTRANDF
    • Computing probability : NTFDIST
    • Computing mean : NTFMEAN
    • Computing standard deviation : NTFSTDEV
    • Computing skewness : NTFSKEW
    • Computing kurtosis : NTFKURT
    • Computing moments above at once : NTFMOM

Reference​