Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate continuous distributions ▷ F-distribution ▷ Definition

Definition: Let $X_1$ and $X_2$ be independent random variables following a chi-squared distribution with $d_1$ and $d_2$ degrees of freedom, respectively:

\[\label{eq:chi2} \begin{split} X_1 &\sim \chi^{2}(d_1) \\ X_2 &\sim \chi^{2}(d_2) \; . \end{split}\]

Then, the ratio of $X_1$ to $X_2$, divided by their respective degrees of freedom, is said to be $F$-distributed with numerator degrees of freedom $d_1$ and denominator degrees of freedom $d_2$:

\[\label{eq:F} Y = \frac{X_1 / d_1}{X_2 / d_2} \sim F(d_1,d_2) \quad \text{where} \quad d_1, d_2 > 0 \; .\]

The $F$-distribution is also called “Snedecor’s $F$-distribution” or “Fisher–Snedecor distribution”, after Ronald A. Fisher and George W. Snedecor.


Metadata: ID: D146 | shortcut: f | author: JoramSoch | date: 2020-04-21, 07:26.