Geometric Distribution: Definition and Probability Calculations | 几何分布的定义与概率计算

📚 Geometric Distribution: Definition and Probability Calculations | 几何分布的定义与概率计算

The geometric distribution is a discrete probability distribution that models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials. It is a key topic in Edexcel A-Level Further Mathematics, often appearing in statistics sections alongside the binomial and Poisson distributions.

几何分布是一种离散概率分布，用于描述在一系列独立的伯努利试验中，首次成功所需的试验次数。它是 Edexcel A-Level 进阶数学中的一个重要考点，常与二项分布和泊松分布一起出现在统计学部分。

1. When Is a Geometric Distribution Used? | 何时使用几何分布？

A geometric distribution is appropriate when we repeat an experiment that has only two possible outcomes — success or failure. The experiment must be repeated until the first success occurs, and each trial must be independent and have the same probability of success.

当我们在重复一个只有两种可能结果（成功或失败）的试验，并且不断重复直到第一次成功出现时，就适用几何分布。每次试验必须相互独立，且成功的概率始终保持不变。

Each trial has exactly two outcomes: success or failure.

每次试验只有两种结果：成功或失败。
The probability of success, denoted p, is the same for every trial.

每次试验成功的概率 p 保持不变。
The trials are independent.

各次试验相互独立。
The random variable X counts the number of trials up to and including the first success.

随机变量 X 表示直到第一次成功（含该次）为止所需的总试验次数。

For example, rolling a fair die until a 6 appears: X = the number of rolls needed. Since the die rolls are independent and each has probability 1/6 of giving a 6, X follows a geometric distribution with p = 1/6.

例如，不断掷一颗均匀骰子直到出现 6：X 表示所需的掷骰次数。由于每次掷骰相互独立且出现 6 的概率均为 1/6，所以 X 服从参数 p = 1/6 的几何分布。

2. Notation and Parameter | 记号与参数

If a random variable X follows a geometric distribution with parameter p, we write X ∼ Geo(p). Here p is the probability of success in a single trial, and it must satisfy 0 < p ≤ 1.

如果随机变量 X 服从参数为 p 的几何分布，我们记作 X ∼ Geo(p)。这里 p 是单次试验中成功的概率，且必须满足 0 < p ≤ 1。

In the Edexcel specification, the geometric distribution is often defined in terms of the number of trials required to obtain the first success, not the number of failures before the first success. This is an important distinction to remember in exams.

在 Edexcel 考纲中，几何分布通常被定义为“获得第一次成功所需的试验次数”，而不是“第一次成功之前的失败次数”。这是考试中需要特别注意的区别。

X ∼ Geo(p)
P(X = x) = p(1 – p)^x-1, x = 1, 2, 3, …

3. Deriving the Probability Mass Function | 概率质量函数的推导

To have the first success on trial x, the first x – 1 trials must all be failures, and the x-th trial must be a success. Since the trials are independent, we multiply the probabilities.

若要在第 x 次试验才首次成功，则前 x – 1 次试验都必须失败，而第 x 次试验必须成功。由于各次试验相互独立，我们将概率相乘即可。

The probability of one failure is 1 – p. Therefore the probability of x – 1 consecutive failures is (1 – p)^x-1. Multiplying by p for the final success gives:

一次失败的概率是 1 – p。因此连续 x – 1 次失败的概率是 (1 – p)^x-1。再乘以最后成功的概率 p，得到：

P(X = x) = (1 – p)^x-1 p

This formula is often written as p(1 – p)^x-1, which is the standard form used in Edexcel textbooks.

这个公式通常写作 p(1 – p)^x-1，也是 Edexcel 教材中使用的标准形式。

4. Verifying That It Is a Valid Probability Distribution | 验证它是合法的概率分布

For any valid probability distribution, the sum of all probabilities must equal 1. For the geometric distribution, the probabilities form a geometric series with first term p and common ratio 1 – p.

对于任何合法的概率分布，所有概率之和必须等于 1。对几何分布而言，这些概率构成一个首项为 p、公比为 1 – p 的等比级数。

∑_x=1^∞ p(1 – p)^x-1 = p + p(1-p) + p(1-p)² + … = p / [1 – (1-p)] = 1

Since the common ratio 1 – p lies between 0 and 1, the infinite series converges, confirming that the total probability is exactly 1.

由于公比 1 – p 介于 0 和 1 之间，该无穷级数收敛，从而验证了总概率恰好等于 1。

5. Cumulative Distribution Function | 累积分布函数

The cumulative distribution function (CDF) gives P(X ≤ x), the probability that the first success occurs on or before trial x. This is often easier to compute using the complement rule.

累积分布函数给出 P(X ≤ x)，即首次成功出现在第 x 次或之前的概率。通常利用补事件法则计算更为方便。

Notice that the event X > x means that the first x trials are all failures. Therefore:

注意到事件 X > x 表示前 x 次试验全部失败。因此：

P(X > x) = (1 – p)^x

Hence the CDF is:

因此累积分布函数为：

P(X ≤ x) = 1 – (1 – p)^x , x = 1, 2, 3, …

This formula is extremely useful for solving problems that require finding P(X < x) or P(X ≥ x).

这个公式在求解涉及 P(X < x) 或 P(X ≥ x) 的问题时非常有用。

6. Mean and Variance | 均值与方差

The expected value, or mean, of a geometric distribution is the reciprocal of p. Intuitively, if the probability of success is 1/6, then on average we expect to wait 6 trials for the first success.

几何分布的期望（均值）等于 p 的倒数。直觉上，如果成功概率是 1/6，那么平均需要等待 6 次试验才会出现首次成功。

E(X) = 1 / p

The variance of a geometric distribution is also expressed in terms of p:

几何分布的方差也用 p 表示：

Var(X) = (1 – p) / p²

These results are given in the Edexcel formula booklet, but you should know how to apply them in context.

这两个结果在 Edexcel 公式册中给出，但你需要掌握如何在具体问题中应用它们。

7. Example: Direct Probability Calculation | 例题：直接计算概率

A biased coin has probability 0.3 of landing heads. The coin is tossed repeatedly until a head appears. Let X be the number of tosses required. Find P(X = 4).

一枚不均匀硬币正面朝上的概率为 0.3。反复抛掷这枚硬币直到出现正面。设 X 为所需抛掷次数。求 P(X = 4)。

Using the PMF with p = 0.3 and x = 4:

使用概率质量函数，p = 0.3，x = 4：

P(X = 4) = 0.3 × (0.7)³ = 0.3 × 0.343 = 0.1029

So the probability that the first head appears on the fourth toss is 0.1029.

因此第一次正面出现在第四次抛掷的概率为 0.1029。

8. Example: Using the CDF | 例题：利用累积分布函数

Using the same biased coin, find the probability that the first head appears within the first 5 tosses.

仍使用上述不均匀硬币，求第一次正面出现在前 5 次抛掷之内的概率。

We need P(X ≤ 5). Using the CDF formula:

我们需要 P(X ≤ 5)。使用累积分布函数公式：

P(X ≤ 5) = 1 – (0.7)⁵ = 1 – 0.16807 = 0.83193

Thus there is about an 83.2% chance that the first head appears within the first five tosses.

因此，第一次正面出现在前五次抛掷之内的概率约为 83.2%。

9. Example: Finding an Unknown Parameter | 例题：求未知参数

Suppose X ∼ Geo(p) and P(X ≤ 3) = 0.875. Find p.

设 X ∼ Geo(p) 且 P(X ≤ 3) = 0.875。求 p。

Using the CDF:

利用累积分布函数：

1 – (1 – p)³ = 0.875
(1 – p)³ = 0.125
1 – p = 0.5
p = 0.5

So the success probability in each trial is 0.5.

因此每次试验的成功概率为 0.5。

10. The Memoryless Property | 无记忆性

A remarkable property of the geometric distribution is that it is memoryless. This means that the probability of waiting at least a further k trials, given that we have already waited n trials, is the same as the probability of waiting at least k trials from the start.

几何分布有一个显著性质——无记忆性。这意味着在已经等待了 n 次之后，还需要至少再等待 k 次的概率，与从头开始需要至少等待 k 次的概率相同。

P(X > n + k | X > n) = P(X > k) = (1 – p)^k

This property is unique to the geometric distribution among the discrete distributions studied in A-Level Further Mathematics.

在 A-Level 进阶数学所学的离散分布中，无记忆性是几何分布独有的性质。

11. Geometric vs. Binomial Distribution | 几何分布与二项分布的区别

Both geometric and binomial distributions involve independent Bernoulli trials, but they answer different questions. A binomial distribution fixes the number of trials and counts the total number of successes. A geometric distribution fixes the number of successes (exactly 1) and counts the number of trials needed to achieve it.

几何分布和二项分布都涉及独立的伯努利试验，但它们回答的问题不同。二项分布固定试验次数，统计成功总次数；几何分布则固定成功次数（恰好 1 次），统计达到该成功所需的试验次数。

Binomial: X = number of successes in n fixed trials.

二项分布：X 为在固定 n 次试验中的成功次数。
Geometric: X = number of trials until the first success.

几何分布：X 为直到第一次成功为止的试验次数。

For example, tossing a coin 10 times and counting heads is binomial. Tossing a coin until the first head appears is geometric.

例如，抛一枚硬币 10 次并统计正面次数属于二项分布；不断抛一枚硬币直到第一次出现正面则属于几何分布。

12. Common Pitfalls and Exam Tips | 常见错误与考试技巧

Students often make errors by confusing the geometric distribution with the negative binomial, or by using the wrong exponent in the PMF. Remember that the exponent is x – 1, not x, because the success occurs on the x-th trial.

学生常犯的错误是将几何分布与负二项分布混淆，或者在概率质量函数中写错指数。记住指数是 x – 1 而不是 x，因为成功发生在第 x 次试验。

Always confirm whether X represents the number of trials or the number of failures before the first success.

始终确认 X 表示的是“试验次数”还是“第一次成功前的失败次数”。
Use the CDF for questions like “at least” or “at most”.

对于“至少”或“至多”类问题，优先使用累积分布函数。
Check that your computed probabilities are between 0 and 1.

检查计算出的概率是否在 0 和 1 之间。
In exam questions, read carefully whether the distribution is Geo(p) or another transformed version.

考试中要仔细阅读题干，确认是 Geo(p) 还是其它变换后的版本。

Mastering the geometric distribution involves recognising the setup, applying the correct formula, and interpreting the results in context. Practice with past papers to build confidence.

掌握几何分布的关键在于识别题型、运用正确的公式，并结合实际情境解释结果。通过练习历年真题来增强信心。

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Geometric Distribution: Definition and Probability Calculations | 几何分布的定义与概率计算