What is F-distribution

F-distribution is a probability distribution that the following random variable F follows when the random variables X and Y follow chi-squared distribution \chi^2(m) and \chi^2(n) respectively and X and Y are independent from each other.

The F distribution is sometimes denoted F(m,n), and the probability density function of the F-distribution is expressed by the following equation:

The graph of the F-distribution depends on the degrees of freedom m and n and looks like this.

Expected value and variance of the F-distribution

The expected value and variance of the random variable X following an F-distribution F(m, n) are respectively:

F-distribution table (upper side)

Since the F-distribution has only m and n parameters, the probabilities of the F-distribution can be summarized in a table called the F-distribution table. Below is the F-distribution table for the upper side probability \alpha=0.05.

For example, the F value with \alpha=0.05 for m=5 and n=2 degrees of freedom is 19.296.

Python Code

The following is the Python code used to draw the F-distribution.

from scipy import stats
import matplotlib.pyplot as plt
import numpy as np

plt.style.use('ggplot')
fig, ax = plt.subplots(facecolor="w", figsize=(10, 5))

x = np.linspace(0.000001, 10, 1000) # X axis
f_df = [(5,5), (5,10), (5,20), (10,5), (20,5), (10,10), (20,20)] # degree of freedom of F(m,n)

for i in range(len(f_df)):
    y = stats.f.pdf(x, f_df[i][0], f_df[i][1])
    ax.plot(x, y, linestyle='-', label=f'F({f_df[i][0]}, {f_df[i][1]})', lw=5, alpha=0.5)

plt.xlim(0, 5)
plt.ylim(0, 1.0)
plt.legend()
plt.show()