# F distribution

## Shape of Distribution

### Basic Properties

• Two parameters $N_1$ and $N_2$ are required (Positive integer)
• Continuous distribution defined on semi-bounded range $x \geq 0$
• This distribution is asymmetric.

### Probability

• Probability density function
$f(x)=\frac{\Gamma\left(\frac{N_1+N_2}{2}\right)\left(\frac{N_1}{N_2}\right)^\frac{N_1}{2}x^{\frac{N_1}{2}-1}}{\Gamma\left(\frac{N_1}{2}\right)\Gamma\left(\frac{N_2}{2}\right)\left(1+\frac{N_1}{N_2}x\right)^{\frac{N_1+N_2}{2}}}$

, where $\Gamma(\cdot)$ is gamma function.

• Cumulative distribution function
$F(x)=I_\gamma\left(\frac{N_1}{2},\frac{N_2}{2}\right)$

, where $\gamma=\frac{N_1x}{N_2+N_1x}$ and $I_{x}(\cdot,\cdot)$ is regularized incomplete beta function.

• How to compute these on Excel.

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A B
Data Description
5 Value for which you want the distribution
4 Value of parameter N1
30 Value of parameter N2
Formula Description (Result)
=NTFDIST(A2,A3,A4,TRUE) Cumulative distribution function for the terms above
=NTFDIST(A2,A3,A4,FALSE) Probability density function for the terms above
• Function reference : NTFDIST

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

• Mean of the distribution is given as
$\frac{N_2}{N_2-2}\quad (N_2>2)$
• How to compute this on Excel

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Data Description
8 Value of parameter N2
Formula Description (Result)
=NTFMEAN(A2) Mean of the distribution for the terms above
• Function reference : NTFMEAN

### Standard Deviation – How wide does the distribution spread? (Definition)

• Variance of the distribution is given as
$\frac{2N_2^2(N_1+N_2-2)}{N_1(N_2-2)^2(N_2-4)}\quad (N_2>4)$

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

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Data Description
4 Value of parameter N1
30 Value of parameter N2
Formula Description (Result)
=NTFSTDEV(A2,A3) Standard deviation of the distribution for the terms above
• Function reference : NTFSTDEV

### Skewness – Which side is the distribution distorted into? (Definition)

• Skewness of the distribution is given as
$\frac{(2N_1+N_2-2)\sqrt{8(N_2-4)}}{\sqrt{N_1(N_1+N_2-2)}(N_2-6)}\quad (N_2>6)$
• How to compute this on Excel

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Data Description
4 Value of parameter N1
30 Value of parameter N2
Formula Description (Result)
=NTFSKEW(A2,A3) Skewness of the distribution for the terms above
• Function reference : NTFSKEW

### Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

• Kurtosis of the distribution is given as
$\frac{12[(N_2-2)^2(N_2-4)+N_1(N_1+N_2-2)(5N_2-22)]}{N_1(N_2-6)(N_2-8)(N_1+N_2-2)}\quad (N_2>8)$
• This distribution can be leptokurtic or platykurtic.
• How to compute this on Excel

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Data Description
4 Value of parameter N1
30 Value of parameter N2
Formula Description (Result)
=NTFKURT(A2,A3) Kurtosis of the distribution for the terms above
• Function reference : NTFKURT

## Random Numbers

• How to generate random numbers on Excel.

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Data Description
4 Value of parameter N1
30 Value of parameter N2
Formula Description (Result)
=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.

• Function reference : NTRANDF

## 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