T.8 · Условие устойчивости, байесовский вывод и ошибка выборки
GRE Subject Test · GRE Mathematics Тема 21 3:14 Английское озвучивание · Английский + китайские субтитры (встроенные)
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This unit separates conditional, joint and independent-event probabilities, calculates posterior probabilities with a complete base-rate table, and uses variance and sample size to find a sample mean's standard error.
本单元区分条件概率、联合概率和独立事件的概率，用完整的基础比率表计算后验概率，并用方差和样本量求样本均值的标准误。
Restrict to B before computing the fraction in A.
先限制在 B 内，再计算其中属于 A 的比例。
Independence means the joint probability is the product.
独立意味着联合概率等于乘积。
With P of A 0.4, P of B 0.5 and joint 0.3, the conditional is 0.6 and the events are not independent.
P(A) = 0.4，P(B) = 0.5，联合概率 0.3 时，条件概率是 0.6，事件不独立。
Mutually exclusive events with positive probabilities cannot be independent.
概率为正的互斥事件不可能独立。
Prevalence 2 percent, sensitivity 90, specificity 95, in a population of 10,000.
患病率 2%，敏感度 90%，特异度 95%，总人数 10,000。
Of 200 with the condition, 180 test positive; of 9,800 without it, 490 test positive.
200 名患者中 180 人阳性；9,800 名非患者中 490 人阳性。
So 180 of 670 positives have the condition: about 27 percent.
所以 670 个阳性中有 180 个确实患病：约 27%。
The 90 percent answers a different question.
90% 回答的是另一个问题。
For independent observations with variance sigma squared, the sample mean has standard error sigma over root n.
对方差为 σ² 的独立观测，样本均值的标准误是 σ/√n。
With standard deviation 12 and n of 36, the mean standard error 2; n of 144 gives 1.
标准差 12、n = 36 时，均值标准误为 2；n = 144 时为 1。
Quadrupling n halves it, and the spread of one observation stays 12.
n 变为 4 倍，标准误减半，而单个观测的离散程度仍是 12。
With P of A 0.4, P of B 0.5 and joint 0.3, P of A given B is 0.6, not 0.4, so A and B are dependent.
P(A) = 0.4，P(B) = 0.5，联合概率 0.3 时，P(A | B) = 0.6，不是 0.4，所以 A 与 B 不独立。
For standard deviation 12, samples of 36 and 144 have mean standard errors 2 and 1.
标准差为 12 时，样本量 36 和 144 的均值标准误分别是 2 和 1。
Do not reverse P of positive given D into P of D given positive.
不要把 P(+ | D) 颠倒成 P(D | +)。
Include false positives in the posterior denominator.
后验概率的分母要包括假阳性。
And the standard deviation of individual observations and the standard error of their mean behave differently with sample size.
单个观测的标准差与其均值的标准误随样本量的变化方式不同。
Expected counts are 100 defective and 900 not.
期望数量是 100 台有缺陷、900 台无缺陷。
Positives number 80 and 45, so 125 in all, and the probability of a defect given a positive is 80 over 125, which is 0.64.
阳性分别有 80 和 45 台，共 125 台，所以阳性条件下有缺陷的概率是 80/125 = 0.64。
The sensitivity 0.8 is the reversed condition.
敏感度 0.8 是颠倒后的条件概率。
Independence gives standard error 5 over 10, which is 0.5.
独立时标准误为 5/10 = 0.5。
With covariance 1 between every pair, the mean's variance is 1 plus 24 over n: 1.24 at n equals 100.
若每对之间协方差为 1，均值的方差是 1 + 24/n：n = 100 时为 1.24。
The standard error tends to one, not zero, so quadrupling n need not halve it.
标准误趋于 1 而不是 0，所以 n 变为 4 倍不一定使它减半。
Check yourself.
自我检测。
In the 2 percent model the posterior is about 27 percent.
在 2% 的模型中后验概率约为 27%。
Exclusive events with positive probabilities are not independent.
概率为正的互斥事件不独立。
Sigma 12 with n 36 gives standard error 2.
σ = 12、n = 36 给出标准误 2。
Quadrupling n halves it for independent data, but correlated data need not shrink to zero.
对独立数据 n 变 4 倍标准误减半，但相关数据的标准误不一定趋于零。