F Distribution (Fisher-Snedecor)

Compact study note.

Summary

The F distribution is the distribution of a ratio of two independent chi-square variables divided by their degrees of freedom. The name refers to Fisher and Snedecor; the old path title is retained only for link stability.^[1]

Prerequisites

Chi-Square Distribution

Definition

If $U \sim χ_{d_{1}}^{2}$ , $V \sim χ_{d_{2}}^{2}$ , and $U, V$ are independent, then $X = (U / d_{1}) / (V / d_{2}) \sim F_{d_{1}, d_{2}}$ .

Notation and Assumptions

$d_{1}$ and $d_{2}$ are numerator and denominator degrees of freedom.

Parameters

$d_{1} > 0$ and $d_{2} > 0$ .

Support

$(0, \infty)$ .

PMF or PDF

$f_{X} (x) = \frac{(d_{1} / d_{2})^{d_{1} / 2} x^{d_{1} / 2 - 1}}{B (d_{1} / 2, d_{2} / 2) (1 + d_{1} x / d_{2})^{(d_{1} + d_{2}) / 2}}$ for $x > 0$ .

CDF

The CDF is computed through the regularized beta function.

Moments

$E [X] = d_{2} / (d_{2} - 2)$ for $d_{2} > 2$ ; $Var (X) = \frac{2 d_{2}^{2} (d_{1} + d_{2} - 2)}{d_{1} (d_{2} - 2)^{2} (d_{2} - 4)}$ for $d_{2} > 4$ .

Essential Result

The F distribution is right-skewed and has no finite upper endpoint.

Small Example

If $U \sim χ_{5}^{2}$ and $V \sim χ_{10}^{2}$ are independent, then $(U / 5) / (V / 10) \sim F_{5, 10}$ .

Common Mistakes

Calling the distribution symmetric or bounded above.
Describing chi-square variables like cubes of normals instead of sums of squared standard normals.

Connections

References

NIST/SEMATECH, e-Handbook of Statistical Methods, "1.3.6.6 Gallery of Distributions", https://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm ↩︎