📚 Geometric and Negative Binomial Distributions | 几何分布与负二项分布
In Edexcel A-Level Mathematics, geometric and negative binomial distributions arise when we count the number of independent Bernoulli trials needed to achieve a specified number of successes. The geometric distribution models the waiting time until the first success, while the negative binomial distribution extends this idea to the r-th success. These distributions are especially useful in quality control, games of chance, and reliability testing, and they appear regularly in Edexcel Statistics and Further Statistics questions.
在 Edexcel A-Level 数学中，当我们需要计算达到指定成功次数所需的独立伯努利试验次数时，就会用到几何分布和负二项分布。几何分布用来描述直到第一次成功所需的等待时间，而负二项分布则将这一思想推广到第 r 次成功。这些分布在质量控制、概率游戏和可靠性检验中非常有用，也经常出现在 Edexcel 统计与进阶统计的考题中。
1. Defining Bernoulli Trials and Waiting-Time Random Variables | 定义伯努利试验与等待时间随机变量
A Bernoulli trial is a single experiment with exactly two possible outcomes: success, which occurs with probability p, and failure, which occurs with probability 1 − p. A sequence of Bernoulli trials is independent if the outcome of any trial does not affect the others, and identical if p remains constant throughout. Waiting-time random variables count the number of trials performed until a success condition is met.
伯努利试验是指只有两种可能结果的单次实验：成功，其发生概率为 p；失败，其发生概率为 1 − p。如果每一次试验的结果不影响其他试验，则称这组伯努利试验是独立的；如果 p 始终保持不变，则称其是相同的。等待时间随机变量计算的是在满足某个成功条件之前所进行的试验次数。
In Edexcel questions, you should first identify whether the trials are independent, whether there are exactly two outcomes, and whether the probability of success is constant. If these conditions hold, a geometric or negative binomial model may be appropriate. Common examples include rolling a die until a six appears, testing components until one fails, or interviewing candidates until a suitable applicant is found.
在 Edexcel 考题中，你应首先判断试验是否独立、是否只有两种结果、以及成功概率是否保持不变。如果这些条件成立，就可以考虑使用几何分布或负二项分布模型。常见例子包括：反复掷骰子直到出现 6、检测元件直到有一个失效、或面试候选人直到找到合适的申请者。
2. Geometric Distribution: Definition and Conditions | 几何分布的定义与条件
The geometric distribution is denoted by X ~ on. The parameter p is the probability of success on any single trial, where 0 < p ≤ 1.
几何分布记作 X ~ Geo(p)。在 Edexcel 的惯例中，X 表示直到第一次成功为止所进行的试验次数，包括成功的那一次。因此 X 的可能取值为 x = 1, 2, 3 等。参数 p 是任意一次试验中成功的概率，满足 0 < p ≤ 1。
ly. Edexcel almost always uses the trials-to-first-success version, so P(X = x) = p(1 − p)x − 1 for x = 1, 2, 3, ...
当重复进行独立伯努利试验直到第一次成功时，适合使用几何分布。如果题目问的是第一次成功之前失败的次数，则需要在两种定义之间小心转换。Edexcel 几乎总是使用“试验次数直到首次成功”的版本，因此 P(X = x) = p(1 − p)x − 1，其中 x = 1, 2, 3, ...
3. Probability Mass Function of the Geometric Distribution | 几何分布的概率质量函数
For X ~ Geo(p), the probability that the first success occurs on the x-th trial is found by requiring x − 1 failures followed by one success. Since trials are independent, these probabilities multiply, giving the probability mass function shown below.
对于 X ~ Geo(p)，第一次成功发生在第 x 次试验的概率可以这样理解：前 x − 1 次都是失败，然后第 x 次成功。由于试验相互独立，这些概率相乘，得到下面的概率质量函数。
P(X = x) = p(1 − p)x − 1 , x = 1, 2, 3, ...
and the probability that the first six occurs on the fourth roll is P(X = 4) = (1/6)(5/6)3 = 125/1296 ≈ 0.0965.
该公式是大多数几何概率计算的关键。例如，如果反复掷一颗均匀骰子直到出现 6，则 p = 1/6，第一次出现 6 在第 4 次掷骰的概率为 P(X = 4) = (1/6)(5/6)3 = 125/1296 ≈ 0.0965。
A common exam mistake is to use px(1 − p)x − 1 or to include an extra factor. Remember: there is exactly one success, and it must be the final trial in the sequence. The exponent of (1 − p) is therefore x − 1, not x.
常见的考试错误是写成 px(1 − p)x − 1 或多加了一个因子。请记住：只有一次成功，而且它必须发生在最后一次试验。因此 (1 − p) 的指数是 x − 1，而不是 x。
4. Cumulative Probabilities and Tail Probabilities for Geo(p) | 几何分布的累积概率与尾部概率
ccurs in the first x trials. Therefore P(X ≤ x) = 1 − P(no success in the first x trials).
对于几何随机变量，累积概率 P(X ≤ x) 可以通过对几何级数求和得到，但结果有一个简单的封闭形式。事件 X ≤ x 表示第一次成功发生在第 x 次试验或之前，这等价于在前 x 次试验中至少有一次成功。因此 P(X ≤ x) = 1 − P(前 x 次试验全部失败)。
P(X ≤ x) = 1 − (1 − p)x
P(X > x) = (1 − p)x
remely useful because they avoid summing many individual probabilities. For instance, if X ~ 2), the probability that the first success takes more than 5 trials is P(X > 5) = (0.8)5 = 0.32768. Equivalently, P(X ≤ 5) = 1 − 0.32768 = 0.67232.
这些公式非常有用，因为它们避免了对许多单独概率的求和。例如，若 X ~ Geo(0.2)，则第一次成功需要超过 5 次试验的概率为 P(X > 5) = (0.8)5 = 0.32768。等价地，P(X ≤ 5) = 1 − 0.32768 = 0.67232。
obability. Use tail probabilities and rearrange: P(X ≤ x) ≥ c is equivalent to 1 − (1 − p)x ≥ c, so (1 − p)x ≤ 1 − c.
在 Edexcel 考试题中，你可能需要求 P(X > a)、P(a < X ≤ b)，或求使成功概率至少达到某个给定值的最小试验次数。使用尾部概率并变形：P(X ≤ x) ≥ c 等价于 1 − (1 − p)x ≥ c，也就是 (1 − p)x ≤ 1 − c。
5. Mean and Variance of Geo(p) | 几何分布的均值与方差
The expected value of a geometric random variable tells us the average number of trials needed to obtain the first success. For X ~ Geo(p), the mean and variance are given by the following standard results, which are provided in the Edexcel formula booklet.
几何随机变量的期望值告诉我们获得第一次成功平均需要进行的试验次数。对于 X ~ Geo(p)，均值和方差由以下标准结果给出，这些结果已在 Edexcel 公式手册中提供。
E(X) = 1 / p
Var(X) = (1 − p) / p²
0625 = 12, which shows that geometric distributions are highly spread out when p is small.
例如，如果成功概率为 p = 0.25，则 E(X) = 1 / 0.25 = 4，所以平均需要 4 次试验才能获得第一次成功。方差为 Var(X) = 0.75 / 0.25² = 0.75 / 0.0625 = 12，这表明当 p 较小时，几何分布的离散程度非常大。
These results can be derived using the moment generating function or by summing infinite series, but for Edexcel A-Level you are mainly expected to apply them. Remember that the mean is always at least 1, and the variance increases rapidly as p decreases.
这些结果可以用矩母函数或无穷级数求和来推导，但在 Edexcel A-Level 中，你主要需要会应用它们。请记住，均值总是至少为 1，而且当 p 减小时方差会迅速增大。
6. The Memoryless Property | 无记忆性
The geometric distribution has a remarkable property called memorylessness. It states that if you have already observed a certain number of failures, the future waiting time for the first success is unaffected by the past. In other words, the distribution 'forgets' how many trials have already occurred.
几何分布具有一个显著的性质，称为无记忆性。它表明，如果你已经观察到一定次数的失败，那么未来等待第一次成功的时间不受过去影响。换句话说，该分布会“忘记”已经发生了多少次试验。
P(X > s + t | X > s) = P(X > t) = (1 − p)t
ill survive at least t more tests is exactly the same as if it were new. For example, if X ~ Geo(0.1), then P(X > 8 | X > 5) = P(X > 3) = (0.9)3 0.729. This property is unique to the geometric distribution among discrete distributions.
这意味着，如果一个元件已经经历了 s 次测试都没有成功，那么它至少还能再经历 t 次测试的概率与一个全新元件完全相同。例如，若 X ~ Geo(0.1)，则 P(X > 8 | X > 5) = P(X > 3) = (0.9)3 = 0.729。在离散分布中，这一性质是几何分布所特有的。
In exam problems, the memoryless property often provides a shortcut. Instead of calculating conditional probabilities using the full definition, you can simply reset the count. This is especially useful when a question states that a process has already failed several times and asks for the probability of further failures.
在考试题中，无记忆性常常提供一种简便方法。你不必使用完整的条件概率定义来计算，而可以直接重新开始计数。当题目说明一个过程已经失败了几次，并要求计算进一步失败的概率时，这一性质尤其有用。
7. Negative Binomial Distribution: Definition and Relation to Geo(p) | 负二项分布的定义及其与几何分布的关系
The negative binomial distribution generalises the geometric distribution from the first success to the r-th success. Let X be the number of trials needed to achieve a fixed number r of successes, where each trial is an independent Bernoulli trial with success probability p. We write X ~ NB(r, p).
负二项分布将几何分布从第一次成功推广到第 r 次成功。设 X 表示达到固定成功次数 r 所需的试验次数，其中每次试验都是独立伯努利试验，成功概率为 p。我们记作 X ~ NB(r, p)。
, r + 1, r + 2, ... This matches the geometric distribution when r = 1: NB(1, p) = Geo(p).
请注意，不同的教材使用不同的约定。有些教材将负二项变量定义为第 r 次成功之前失败的次数。Edexcel 的题目通常将 X 定义为总试验次数，因此可能取值从 x = r, r + 1, r + 2, ... 开始。当 r = 1 时，这与几何分布一致：NB(1, p) = Geo(p)。
Since the r-th success occurs on trial x exactly when there are r − 1 successes in the first x − 1 trials, followed by a success on trial x, the negative binomial probability can be built from binomial probabilities. This relationship also shows that a negative binomial random variable is the sum of r independent geometric random variables.
由于第 r 次成功发生在第 x 次试验，当且仅当前 x − 1 次试验中有 r − 1 次成功，接着第 x 次试验成功，因此负二项概率可以由二项概率构造出来。这个关系也表明，负二项随机变量是 r 个独立几何随机变量之和。
8. Probability Mass Function of NB(r, p) | 负二项分布的概率质量函数
For X ~ NB(r, p), the probability that the r-th success occurs on the x-th trial is given by the probability mass function below. Here C(n, k) denotes the binomial coefficient, equal to n! / [k!(n − k)!].
对于 X ~ NB(r, p)，第 r 次成功发生在第 x 次试验的概率由下面的概率质量函数给出。其中 C(n, k) 表示二项系数，等于 n! / [k!(n − k)!]。
P(X = x) = C(x − 1, r − 1) pr (1 − p)x − r , x = r, r + 1, r + 2, ...
The binomial coefficient C(x − 1, r − 1) counts the number of ways to arrange the r − 1 successes among the first x − 1 trials. The factor pr represents the r successes, and (1 − p)x − r represents the x − r failures. The final trial must always be a success, which is why we only arrange successes among the first x − 1 trials.
二项系数 C(x − 1, r − 1) 计算的是在前 x − 1 次试验中安排 r − 1 次成功的方式数。因子 pr 代表 r 次成功，而 (1 − p)x − r 代表 x − r 次失败。最后一次试验必须成功，这就是为什么我们只在前 x − 1 次试验中安排成功的原因。
. The probability that the third success occurs on the 5th shot is P(X = 5) = C(4, 2)(0.7)3(0.3)2 = 6 × 0.343 × 0.09 = 0.18522.
例如，假设一名篮球运动员罚球命中率为 0.7，并且一直投篮直到投中 3 次。第 3 次成功发生在第 5 次投篮的概率为 P(X = 5) = C(4, 2)(0.7)3(0.3)2 = 6 × 0.343 × 0.09 = 0.18522。
binomial coefficients by hand. Always check that x is at least r; otherwise P(X = x) = 0.
当 r 较大时，这些计算会比较繁琐。Edexcel 考试题通常关注较小的 r 值，例如 2 或 3，因此你可以手工计算二项系数。务必检查 x 是否至少为 r；否则 P(X = x) = 0。
9. Mean and Variance of NB(r, p) | 负二项分布的均值与方差
Because a negative binomial random variable with parameters r and p can be written as the sum of r independent geometric random variables, its mean and variance follow directly from the geometric results. This is a very useful derivation and is often examined in Edexcel Further Statistics 1.
由于参数为 r 和 p 的负二项随机变量可以写成 r 个独立几何随机变量之和，它的均值和方差可以直接由几何分布的结果推出。这是一个非常有用的推导，常在 Edexcel 进阶统计 1 中考查。
E(X) = r / p
Var(X) = r(1 − p) / p²
For instance, if X ~ 5. On average, 7.5 trials are needed to achieve 3 successes when each trial succeeds with probability 0.4.
例如，若 X ~ NB(3, 0.4)，则 E(X) = 3 / 0.4 = 7.5，Var(X) = 3 × 0.6 / 0.4² = 1.8 / 0.16 = 11.25。当每次试验成功概率为 0.4 时，平均需要 7.5 次试验才能获得 3 次成功。
te which parameters you are using, because if the alternative convention counts failures, the mean and variance formulas would differ by a constant r.
当 r = 1 时，这些公式就退化为几何分布的均值和方差：E(X) = 1/p，Var(X) = (1 − p)/p²。始终要说明你所使用的参数含义，因为如果采用另一种约定来计算失败次数，均值和方差公式会相差常数 r。
10. Choosing the Correct Distribution and Exam Tips | 选择正确的分布与考试技巧
In Edexcel questions, the wording is the most important clue. If the process stops after the first success, use Geo(p). If the process stops after the r-th success, use NB(r, p). Look for phrases such as 'until the first success', 'before the first success', 'to obtain r successes', or 'the r-th success'.
在 Edexcel 的考题中，措辞是最重要的线索。如果过程在第一次成功后就停止，则使用 Geo(p)。如果过程在第 r 次成功后停止，则使用 NB(r, p)。注意诸如“直到第一次成功”、“在第一次成功之前”、“获得 r 次成功”或“第 r 次成功”等短语。
The table below compares the two distributions under the Edexcel convention of counting the total number of trials.
下表在 Edexcel 计算总试验次数的约定下对这两种分布进行了比较。
| Distribution | Probability mass function | Mean | Variance |
| Geo(p) | p(1 − p)x − 1 | 1/p | (1 − p)/p² |
| NB(r, p) | C(x − 1, r − 1) pr(1 − p)x − r | r/p | r(1 − p)/p² |
obability, check that it lies between 0 and 1. For cumulative geometric probabilities, use the closed-form tail formula rather than summing many terms.
解题时，首先确定 p 和 r，然后写出 X 的取值范围。对于几何分布，x ≥ 1；对于负二项分布，x ≥ r。计算出概率后，检查它是否在 0 到 1 之间。对于几何累积概率，使用封闭形式的尾部公式
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