Geometric Distribution | 几何分布

📚 Geometric Distribution | 几何分布

The geometric distribution models the number of independent Bernoulli trials needed to achieve the first success. It is a fundamental discrete probability distribution widely applied in fields such as quality control, risk analysis, and game theory. Understanding its properties, including its memoryless nature and simple moment formulas, is essential for solving real-world problems involving waiting times.

几何分布描述的是在一系列独立伯努利试验中，获得第一次成功所需的试验次数。它是离散概率分布的基础，广泛应用于质量控制、风险分析和博弈论等领域。理解其性质（包括无记忆性和简单的矩公式）对于解决涉及等待时间的实际问题至关重要。

1. Definition and Basic Setup | 定义与基本设定

A geometric distribution arises when we repeat independent trials each with success probability p, and we count the number of trials X until the first success occurs. The trial outcomes are binary: success (S) or failure (F). The probability of success remains constant across trials, and trials are independent.

当我们重复进行每次成功概率为 p 的独立试验，并统计直到第一次成功出现所需的试验次数 X 时，就产生了几何分布。试验结果只有两种：成功 (S) 或失败 (F)。每次试验的成功概率保持不变，并且各次试验相互独立。

The random variable X can take values 1, 2, 3, … . This definition is sometimes called the “first-success” version. Some textbooks define Y as the number of failures before the first success, giving Y = X − 1 with support 0, 1, 2, …; IB typically uses the “number of trials” version.

随机变量 X 可取 1, 2, 3, … 等值。此定义有时称为“首次成功”版本。有些教材将 Y 定义为首次成功之前的失败次数，即 Y = X − 1，取值范围为 0, 1, 2, …；IB 课程通常使用“试验次数”版本。

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2. Conditions for a Geometric Distribution | 几何分布的条件

To apply a geometric model, four conditions must hold: (1) each trial has exactly two possible outcomes, (2) the probability of success p is the same for every trial, (3) the trials are independent, (4) the variable of interest is the number of trials up to and including the first success.

要应用几何分布模型，需满足四个条件：(1) 每次试验只有两种可能结果；(2) 每次试验的成功概率 p 相同；(3) 试验相互独立；(4) 我们关心的随机变量是直到第一次成功为止（含那次成功）的试验次数。

If we are counting the number of failures before the first success, the support shifts to {0, 1, 2, …} and the PMF becomes P(Y = y) = (1 − p)ʸ p, but the IB syllabus focuses on the number of trials.

如果计数的是首次成功之前的失败次数，取值范围变为 {0, 1, 2, …}，概率质量函数变为 P(Y = y) = (1 − p)ʸ p，但 IB 大纲侧重于试验次数版本。

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3. Probability Mass Function (PMF) | 概率质量函数

P(X = k) = (1 − p)ᵏ⁻¹ p, for k = 1, 2, 3, …

对于 X ~ Geo(p)，概率质量函数为 P(X = k) = (1 − p)ᵏ⁻¹ p，其中 k = 1,2,3,…。

This formula reflects the need for k−1 consecutive failures (each with probability 1−p) followed by one success (with probability p). The sum of all probabilities over the entire support equals 1 because ∑_{k=1}^{∞} (1−p)ᵏ⁻¹ p = p / (1 − (1−p)) = 1, using the geometric series formula.

该公式反映了需要先出现 k−1 次连续失败（每次概率为 1−p），然后出现一次成功（概率为 p）。整个取值范围内所有概率之和等于 1，因为 ∑_{k=1}^{∞} (1−p)ᵏ⁻¹ p = p / (1 − (1−p)) = 1，利用了等比级数求和公式。

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4. Cumulative Distribution Function (CDF) | 累积分布函数

F(k) = P(X ≤ k) = 1 − (1 − p)ᵏ, for k = 1, 2, 3, …

累积分布函数为 F(k) = P(X ≤ k) = 1 − (1 − p)ᵏ，其中 k = 1,2,3,…。

The cumulative probability can be derived as P(X ≤ k) = 1 − P(X > k) = 1 − (1 − p)ᵏ, because the event X > k requires that the first k trials are all failures. For example, if p = 0.2, the probability that the first success occurs within the first 3 trials is P(X ≤ 3) = 1 − (0.8)³ = 1 − 0.512 = 0.488.

累积概率可推导为 P(X ≤ k) = 1 − P(X > k) = 1 − (1 − p)ᵏ，因为事件 X > k 要求前 k 次试验全部失败。例如，如果 p = 0.2，首次成功发生在前 3 次试验内的概率为 P(X ≤ 3) = 1 − (0.8)³ = 1 − 0.512 = 0.488。

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5. Expected Value (Mean) | 期望值（均值）

E(X) = 1 / p

期望值为 E(X) = 1 / p。

Intuitively, if each trial has success probability p, the average waiting time is the reciprocal of p. For instance, if p = 0.1, one expects 10 trials on average. A formal derivation uses the definition E(X) = ∑_{k=1}^{∞} k·(1−p)ᵏ⁻¹ p = 1/p, applying the sum of an arithmetico-geometric series.

直观上，如果每次试验的成功概率为 p，平均等待时间为 p 的倒数。例如，若 p = 0.1，则平均需要 10 次试验。正式推导需利用定义 E(X) = ∑_{k=1}^{∞} k·(1−p)ᵏ⁻¹ p = 1/p，其中运用了算术-几何级数的求和公式。

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6. Variance | 方差

Var(X) = (1 − p) / p²

方差为 Var(X) = (1 − p) / p²。

This indicates that the spread of the distribution decreases as p increases. When p is small, the variance is large, reflecting a long right tail. The standard deviation is the square root of the variance, σ = √(1 − p) / p. In calculations, E(X²) is found first, and Var(X) = E(X²) − [E(X)]².

这表明分布的离散程度随着 p 的增大而减小。当 p 很小的时候，方差很大，体现出长长的右尾。标准差为方差的平方根，即 σ = √(1 − p) / p。在计算中，先求出 E(X²)，再使用公式 Var(X) = E(X²) − [E(X)]²。

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7. Memoryless Property | 无记忆性

P(X > m + n | X > m) = P(X > n), for nonnegative integers m, n

无记忆性：对于非负整数 m 和 n，有 P(X > m + n | X > m) = P(X > n)。

This means that, given that no success has occurred in the first m trials, the additional waiting time beyond m has the same distribution as the original waiting time. In practice, it says that past failures do not influence future trials — the process “forgets” its history. This property is due to the independence of trials and constant p.

这意味着，已知前 m 次试验中没有成功，在此条件下还需要再多等待的次数与原始的等待时间服从相同的分布。在实际中，这说明过去的失败并不影响未来的试验——该过程会“遗忘”其历史。这一特性源于试验的独立性和概率 p 的恒定。

Proof: Since P(X > k) = (1 − p)ᵏ, the conditional probability is P(X > m

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