F Distribution — $ F_{\nu_1,\nu_2} $
The F distribution arises as a ratio of two independent Chi-square variables scaled by their degrees of freedom. It is the central distribution for variance ratios, ANOVA, and likelihood ratio tests in linear models.
Definition
Degrees of freedom:
- Numerator: $ \nu_1 \in \mathbb{N} $
- Denominator: $ \nu_2 \in \mathbb{N} $
Support:
\(x \ge 0\)
PDF:
\(f(x)
=
\frac{1}{B(\nu_1/2,\nu_2/2)}
\left(\frac{\nu_1}{\nu_2}\right)^{\nu_1/2}
\frac{x^{\nu_1/2-1}}
{\left(1+\frac{\nu_1}{\nu_2}x\right)^{(\nu_1+\nu_2)/2}}\)
Construction from Chi-square
Let \(U_1 \sim \chi^2_{\nu_1}, \quad U_2 \sim \chi^2_{\nu_2}, \quad U_1 \perp U_2.\)
Then \(F = \frac{(U_1/\nu_1)}{(U_2/\nu_2)} \sim F_{\nu_1,\nu_2}\)
This representation is the definition used in proofs.
Relationship to Other Distributions
Connection to Chi-square
- Ratio of scaled Chi-square variables
- Degrees of freedom come from independent quadratic forms
Connection to t Distribution
If \(T \sim t_\nu,\) then \(T^2 \sim F_{1,\nu}\)
Connection to Beta
If \(X \sim F_{\nu_1,\nu_2},\) then \(\frac{\nu_1 X}{\nu_1 X + \nu_2} \sim \mathrm{Beta}\!\left(\frac{\nu_1}{2},\frac{\nu_2}{2}\right)\)
Moments
-
Mean: \(\mathbb{E}[X] = \frac{\nu_2}{\nu_2-2}, \quad \nu_2>2\)
-
Variance: \(\operatorname{Var}(X) = \frac{2\nu_2^2(\nu_1+\nu_2-2)} {\nu_1(\nu_2-2)^2(\nu_2-4)}, \quad \nu_2>4\)
Higher moments exist only when degrees of freedom are sufficiently large.
Symmetry Property
\[X \sim F_{\nu_1,\nu_2} \quad \Longrightarrow \quad \frac{1}{X} \sim F_{\nu_2,\nu_1}\]Pivotal Quantity (Key Use)
If \(S_1^2, S_2^2\) are independent sample variances from Normal populations, then \(\frac{S_1^2/\sigma_1^2}{S_2^2/\sigma_2^2} \sim F_{n_1-1,n_2-1}\)
This is a pivot for comparing variances.
Role in Classical Inference
Used for:
- ANOVA
- Testing equality of variances
- Regression model comparison
- Likelihood ratio tests in linear models
Likelihood Ratio Tests
In many regular parametric models, \(-2\log\Lambda \Rightarrow \chi^2_k \quad \text{or} \quad F\) depending on normalization.
In linear models, LRTs reduce exactly to F tests.
Tail Behavior
- Right-skewed
- Heavy tails when degrees of freedom are small
- Concentrates near 1 as $ \nu_1,\nu_2 \to \infty $
Multivariate Interpretation (Recognition Only)
- Ratio of independent quadratic forms
- Orthogonal projections in linear regression
(No formulas required for prelims.)
Key Theorems and Facts (Prelim-Relevant)
- Ratio of Chi-square variables
- Square of t is F
- Variance-ratio pivot
- ANOVA and regression tests
- Reciprocal symmetry
Exam Takeaways
- Ratio of variances ⇒ F
- Degrees of freedom come from sample sizes
- $ t^2 $ gives F
- Heavy tails for small df
- Central to linear-model inference
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