The F distribution is a continuous probability distribution which is commonly used in statistical testing procedures.
The F distribution is a continuous probability distribution which is commonly used in statistical testing procedures. If $X$ and $Y$ are two independent identically distributed chi-square random variables with $m$ and $n$ degrees of freedom respectively, then the distribution of the ratio $$F_{m,n} = \frac{(\frac{X}{m})}{(\frac{Y}{n})}$$ is called the F distribution. It is also known as the variance-ratio distribution for the fact that when considering two random samples of size $n$ that follow a normal distribution, the ratio of the two sample variances $s_1^2/s_2^2$ follows an $F_{n_1-1,n_2-1}$ distribution.
The F distribution is frequently used in the analysis of variance (see the F-test tag).
F distribution is for a non-negative random variable. It is skewed to the right. The probability density function is given by: $$f(x|m,n) = \frac{\Gamma (\frac{m+n}{2})}{\Gamma (\frac{m}{2}) \Gamma (\frac{n}{2}}) \cdot \frac{(\frac{m}{2})^{m/2} x^{m/(2-1)}}{(\frac{n}{2})^{m/2} [1+\frac{mx}{n}]^{(m/2)+(n/2)}}$$ And its first two moments are $$\frac{n}{n-2}$$ for the mean and $$\frac{2n^2(m+n-2)}{m(n-2)^2 (n-4)}$$ for the variance (with $n>4$).
