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

Definition: Let $X_1$ and $X_2$ be independent random variables following a chi-squared distribution with $\nu_1$ and $\nu_2$ degrees of freedom, respectively:

\[\label{eq:chi2} \begin{split} X_1 &\sim \chi^{2}(\nu_1) \\ X_2 &\sim \chi^{2}(\nu_2) \; . \end{split}\]

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 $\nu_1$ and denominator degrees of freedom $\nu_2$:

\[\label{eq:F} Y = \frac{X_1 / \nu_1}{X_2 / \nu_2} \sim F(\nu_1, \nu_2) \quad \text{where} \quad \nu_1, \nu_2 > 0 \; .\]

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

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