📚 F Distribution and Its Properties | F分布及其性质
The F distribution, also called the Fisher-Snedecor distribution, is a continuous probability distribution that arises when comparing variances. It is widely used in analysis of variance, regression analysis, and tests for equality of two population variances.
F分布，又称费希尔-斯内德克分布，是一种在比较方差时产生的连续概率分布。它广泛用于方差分析、回归分析以及两个总体方差是否相等的检验。
1. Definition and Notation | 定义与记号
Let U and V be independent random variables following chi-square distributions with degrees of freedom d₁ and d₂ respectively. Then the random variable F = (U/d₁) / (V/d₂) is said to follow an F distribution with d₁ and d₂ degrees of freedom, written as F ~ F(d₁, d₂). The first degree of freedom is the numerator degree, and the second is the denominator degree.
设 U 和 V 分别是自由度为 d₁ 和 d₂ 的独立卡方分布随机变量。则随机变量 F = (U/d₁) / (V/d₂) 称为服从自由度为 d₁ 和 d₂ 的 F 分布，记作 F ~ F(d₁, d₂)。第一个自由度为分子自由度，第二个为分母自由度。
F = (U/d₁) / (V/d₂) ~ F(d₁, d₂)
2. Probability Density Function | 概率密度函数
The probability density function (pdf) of the F distribution is defined for x > 0. It can be written using the Beta function B(a,b). In the A-Level course you are not expected to reproduce the pdf, but you should know its general form and that the area under the curve equals 1.
F 分布的概率密度函数在 x > 0 上有定义。它可以借助 Beta 函数 B(a,b) 来表示。在 A-Level 课程中并不要求默写该表达式，但应当了解其大致形式，并知道曲线下总面积为 1。
f(x) = [ (d₁/d₂)^(d₁/2) · x^(d₁/2 − 1) ] / [ B(d₁/2, d₂/2) · (1 + (d₁/d₂)x)^((d₁+d₂)/2) ], x ≥ 0
3. Non-negativity and Shape | 非负性与形状
Because F is defined as a ratio of two positive variables, its support is all positive real numbers. The distribution is unimodal and positively skewed, with a long right tail. The exact shape depends strongly on the degrees of freedom: for small d₁ and d₂ the curve is heavily skewed, while for large degrees of freedom it becomes more symmetric.
由于 F 是两个正量之比，其取值只可能为正实数。该分布为单峰、右偏，具有长右尾。分布形状强烈依赖自由度：当 d₁ 和 d₂ 较小时曲线严重偏斜；当自由度增大时，分布趋于对称。
4. Expected Value and Variance | 期望与方差
If F follows F(d₁, d₂), its mean exists only when d₂ > 2, and its variance exists only when d₂ > 4. These moments are given by the following formulae.
若 F 服从 F(d₁, d₂)，其期望仅在 d₂ > 2 时存在，方差仅在 d₂ > 4 时存在。
E(F) = d₂ / (d₂ − 2), d₂ > 2
Var(F) = [ 2 d₂² (d₁ + d₂ − 2) ] / [ d₁ (d₂ − 2)² (d₂ − 4) ], d₂ > 4
5. Relationship with the Chi-Square Distribution | 与卡方分布的关系
By definition, F is the ratio of two independent scaled chi-square variables. Conversely, if X ~ F(d₁, d₂) and d₂ is large, then d₁ · X is approximately chi-square distributed with d₁ degrees of freedom. This approximation is occasionally used in statistical inference.
由定义可知，F 是两个独立缩放卡方变量之比。反过来，若 X ~ F(d₁, d₂) 且 d₂ 很大，则 d₁ · X 近似服从自由度为 d₁ 的卡方分布。这一近似在统计推断中偶尔使用。
6. Reciprocal Property | 倒数性质
A very important property is that the reciprocal of an F variable also follows an F distribution with the degrees of freedom swapped. This property is used to obtain lower-tail critical values from an upper-tail table.
一个非常重要的性质是：F 变量的倒数仍然服从 F 分布，但两个自由度互换。这一性质被用于从右尾表中获取左尾临界值。
X ~ F(d₁, d₂) ⇔ 1/X ~ F(d₂, d₁)
7. Relationship with the t-Distribution | 与t分布的关系
If T follows Student’s t distribution with n degrees of freedom, then the square of T follows an F distribution with 1 and n degrees of freedom. This relationship means that a two-tailed t-test can be viewed as an F-test on the squared test statistic.
若 T 服从自由度为 n 的 Student t 分布，则 T 的平方服从自由度为 1 和 n 的 F 分布。这一关系表明，双尾 t 检验可以看作是对检验统计量平方的 F 检验。
T ~ t(n) ⇒ T² ~ F(1, n)
8. Using F Distribution Tables | 查表方法
F tables usually give the upper-tail critical value F_α(d₁, d₂) such that P(F > F_α) = α. To find a critical value, locate the row for the numerator degree of freedom d₁ and the column for the denominator degree of freedom d₂. Always check the table arrangement, since some tables list d₁ along the top. For a two-tailed test, the upper critical value is F_(α/2), and the lower critical value is the reciprocal of F_(α/2) with the degrees of freedom swapped.
F 分布表通常给出右尾临界值 F_α(d₁, d₂)，使得 P(F > F_α) = α。查表时，在行中找到分子自由度 d₁，在列中找到分母自由度 d₂。由于有些表格将 d₁ 排列在表头，务必检查表格的排列方式。对于双尾检验，上临界值为 F_(α/2)，下临界值则取 F_(α/2) 的倒数并将自由度互换。
9. Hypothesis Test for Equality of Two Variances | 两方差相等的假设检验
Assume two independent random samples are drawn from two normal populations with variances σ₁² and σ₂². To test H₀: σ₁² = σ₂² against a one-sided or two-sided alternative, we use the statistic F = s₁² / s₂², where s₁² and s₂² are the sample variances and n₁ − 1, n₂ − 1 are the corresponding degrees of freedom. Under H₀, F follows F(n₁ − 1, n₂ − 1).
设两个独立随机样本分别来自方差为 σ₁² 和 σ₂² 的正态总体。要检验 H₀: σ₁² = σ₂² 对单尾或双尾备择假设，可使用统计量 F = s₁² / s₂²，其中 s₁² 和 s₂² 是样本方差，n₁ − 1 和 n₂ − 1 是相应自由度。在 H₀ 成立时，F ~ F(n₁ − 1, n₂ − 1)。
10. Worked Example | 实例
A manufacturer wants to test whether the variance of machine A’s output is greater than that of machine B. Two independent samples of size 10 give sample variances s_A² = 0.45 and s_B² = 0.21. Test at the 5% significance level.
某制造商想检验机器 A 输出的方差是否大于机器 B。两个独立样本容量均为 10，样本方差分别为 s_A² = 0.45 和 s_B² = 0.21。在 5% 显著性水平下进行检验。
The hypotheses are H₀: σ_A² = σ_B² and H₁: σ_A² > σ_B². The test statistic is F = 0.45 / 0.21 = 2.14. The degrees of freedom are both 9. From the F table, the 5% critical value is F₀.₀₅(9, 9) = 3.18. Since 2.14 < 3.18, we do not reject H₀. There is insufficient evidence to claim that machine A has greater variance.
原假设为 H₀: σ_A² = σ_B²，备择假设为 H₁: σ_A² > σ_B²。检验统计量为 F = 0.45 / 0.21 = 2.14。两个自由度均为 9。查 F 分布表，5% 临界值为 F₀.₀₅(9, 9) = 3.18。因为 2.14 < 3.18，所以不拒绝原假设。没有足够证据表明机器 A 的方差更大。
11. Cautions | 注意事项
The F test for equality of variances is sensitive to the assumption of normality; it should not be used when the populations are clearly non-normal. The order of degrees of freedom matters: F(d₁, d₂) is not the same as F(d₂, d₁). When using tables, verify whether the values are upper-tail or lower-tail probabilities. For two-tailed tests, remember to halve the significance level before using the table.
方差相等性的 F 检验对正态性假设较为敏感；当总体明显非正态时不应使用。自由度的顺序很重要：F(d₁, d₂) 与 F(d₂, d₁) 不同。查表时，需确认表中的数值是右尾概率还是左尾概率。对于双尾检验，使用表格前应将显著性水平减半。
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