F Distribution

Definition

Concept

F distribution: continuous probability distribution of ratio of two scaled chi-square variables. Used to compare variances between samples.

Formula

Defined as ratio: (U1 / d1) / (U2 / d2), where U1 ~ χ²(d1), U2 ~ χ²(d2), independent.

Support

Range: x ∈ [0, ∞). Values non-negative, skewed right.

Historical Development

Origin

Introduced by Ronald A. Fisher (1924) for variance ratio tests.

Early Applications

Developed for analysis of variance (ANOVA) and experimental design.

Evolution

Extended use in regression, model comparison, and inferential statistics.

Mathematical Properties

Parameters

Two degrees of freedom: numerator (d1), denominator (d2).

Shape

Skewed distribution, shape varies with d1 and d2.

Support and Moments

Defined on positive real line; moments exist if d2 > certain thresholds.

Probability Density Function (PDF)

General Form

PDF expressed using beta function and gamma functions.

Formula

f(x; d1, d2) = [ (d1/d2)^(d1/2) * x^(d1/2 - 1) ] / [ B(d1/2, d2/2) * (1 + (d1/d2)*x)^{(d1 + d2)/2} ], x > 0 

Components

B(a,b): Beta function; Γ(z): Gamma function; x: random variable.

Cumulative Distribution Function (CDF)

Definition

CDF: probability that F variable ≤ value x.

Expression

Related to incomplete beta function I.

F(x; d1, d2) = I_{ (d1 x) / (d1 x + d2) } (d1/2, d2/2) 

Properties

Monotonically increasing; approaches 1 as x → ∞.

Moments

Mean

Exists if d2 > 2; mean = d2 / (d2 - 2)

Variance

Exists if d2 > 4; variance = [2 d2² (d1 + d2 - 2)] / [d1 (d2 - 2)² (d2 - 4)]

Higher Moments

Skewness and kurtosis defined for higher d2 values.

Statistical Uses

ANOVA

Test equality of multiple population variances via mean squares ratio.

Regression Analysis

Compare nested models, test overall regression significance.

Hypothesis Testing

Variance ratio tests, model comparison, goodness-of-fit evaluations.

Parameter Interpretation

Degrees of Freedom (d1)

Numerator df: related to variance estimate numerator, usually number of groups minus one.

Degrees of Freedom (d2)

Denominator df: associated with variance estimate denominator, typically total sample size minus number of groups.

Impact on Distribution

Higher df → distribution approaches normality; lower df → skewed, heavy-tailed.

Relationship with Other Distributions

Chi-Square Distribution

F is ratio of two scaled independent chi-square variables.

Beta Distribution

Transformed F-distribution relates to Beta distribution via variable change.

T Distribution

Square of t-distributed variable with d degrees of freedom equals F(1, d).

Critical Values and Tables

Purpose

Critical values determine rejection regions for hypothesis tests.

Table Structure

Indexed by numerator and denominator degrees of freedom and significance levels.

Example Values

Simulation and Sampling

Generating F-Distributed Variables

Generate U1 ~ χ²(d1), U2 ~ χ²(d2), then compute F = (U1/d1) / (U2/d2).

Monte Carlo Methods

Simulate sampling distributions to estimate power and critical values.

Applications

Bootstrap methods, permutation tests, and variance component estimation.

Limitations and Assumptions

Independence

Assumes numerator and denominator chi-square variables are independent.

Normality

Underlying data assumed normally distributed for variance ratio tests.

Sample Size

Small sample sizes distort approximation; degrees of freedom impact accuracy.

References

• Fisher, R.A., "On the 'probable error' of a coefficient of correlation deduced from a small sample," Metron, vol. 1, 1921, pp. 3–32.
• Rao, C.R., "Linear Statistical Inference and Its Applications," Wiley, 1973, pp. 150-160.
• Johnson, N.L., Kotz, S., Balakrishnan, N., "Continuous Univariate Distributions, Volume 2," Wiley-Interscience, 1995, pp. 267-275.
• Casella, G., Berger, R.L., "Statistical Inference," 2nd Ed., Duxbury, 2002, pp. 317-320.
• Hogg, R.V., Craig, A.T., "Introduction to Mathematical Statistics," 7th Ed., Pearson, 2013, pp. 405-410.