F-Distribution

F-distribution is a continuous statistical distribution whose definition is as follows:

where χ²_m and χ²_n are independent variables from chi-square distribution with m and n degrees of freedom, respectively.

By definition, the F-distribution can be expressed as follows:

where each Z_i is an independent variable from the standard normal distribution.

According to the degrees of freedom, the distribution functions have different shape.

Example

What is the probability for random variable x < 1.24 in the F-distribution with the degrees of freedom of d1 = 12 and d2 = 10?

The following will be the input:

• Random variable x = 1.24
• Degree of freedom d1 = 12
• Degree of freedom d2 = 10

The following will be the result of Statistics Study:

F DISTRIBUTION

Your Input:

  Random variable x = 1.24
  Significance level =
  Degree of freedom d1 = 12
  Degree of freedom d2 = 10


Probability Density Funcdtion (PDF)

   x    f(x)
 -------------+-------------------+
  0.0     NaN |
  0.2  0.1130 |***
  0.4  0.5163 |**************
  0.6  0.7506 |********************
  0.8  0.7518 |********************
  1.0  0.6439 |*****************
  1.2  0.5130 |**************
  1.4  0.3950 |***********
  1.6  0.2998 |********
  1.8  0.2266 |******
  2.0  0.1715 |*****
  2.2  0.1305 |***
  2.4  0.0999 |***
  2.6  0.0770 |**
  2.8  0.0598 |**
  3.0  0.0469 |*
  3.2  0.0370 |*
  3.4  0.0294 |*
  3.6  0.0235 |*
  3.8  0.0190 |*
  4.0  0.0154 |
 -------------+-------------------+

  f(1.24, 12, 10) = 0.4878


Cumulative Distribution Function (CDF)

   x    F(x)
 -------------+-------------------+
  0.0  0.0000 |
  0.2  0.0054 |
  0.4  0.0677 |*
  0.6  0.1990 |****
  0.8  0.3521 |*******
  1.0  0.4926 |**********
  1.2  0.6082 |************
  1.4  0.6987 |**************
  1.6  0.7678 |***************
  1.8  0.8201 |****************
  2.0  0.8596 |*****************
  2.2  0.8896 |******************
  2.4  0.9125 |******************
  2.6  0.9301 |*******************
  2.8  0.9437 |*******************
  3.0  0.9543 |*******************
  3.2  0.9626 |*******************
  3.4  0.9692 |*******************
  3.6  0.9745 |*******************
  3.8  0.9787 |********************
  4.0  0.9822 |********************
 -------------+-------------------+

  F(1.24, 12, 10) = P( < 1.24) = 0.6283


|     ___       v1 = 12
|   --   ---    v2 = 10
|  -        ----
| -           | ----
|-            |     ---_____
+-------------+--------------
0             F
    F = 1.24

  P(F < 1.24) = 0.628257
  P(F > 1.24) = 0.371743


Properties of the PDF

  Mean = d2/(d2-2) = 1.25

  Mode = (d1-2)/d1 · d2/(d2+2) = 0.6944

  Variance = 0.8681

  Skewness = 3.5777

  Ex. kurtosis = 44.4