Poisson Cumulative Distribution Function | 泊松累积分布函数

📚 Poisson Cumulative Distribution Function | 泊松累积分布函数

The Poisson distribution is one of the most useful discrete probability models in A‑Level Mathematics. In many real‑world scenarios – such as counting the number of emails arriving in an hour or the number of flaws along a length of wire – we are interested not just in the probability of a single count, but in the cumulative probability of observing up to a certain number of events. This is where the Poisson cumulative distribution function (CDF) becomes essential. In this article we will explore the Poisson CDF in depth, with clear derivations, practical calculation methods, and plenty of examples aligned to the Edexcel specification.

泊松分布是A‑Level数学中最有用的离散概率模型之一。在许多现实情境中——例如统计一小时内收到的电子邮件数量或一段金属线上的瑕疵数目——我们感兴趣的不仅仅是某个单一计数的概率，更包括观察到不超过某一数量事件的累积概率。这正是泊松累积分布函数（CDF）的核心用途。本文将深入探讨泊松CDF，提供清晰的推导、实用的计算方法以及大量符合Edexcel考纲的例题。

1. The Poisson Distribution | 泊松分布简介

Recall that a discrete random variable X follows a Poisson distribution with parameter λ (λ > 0) if it counts the number of events occurring in a fixed interval of time or space, under the assumption that events happen randomly, independently, and at a constant average rate λ.

回顾一下，若离散随机变量X表示在固定时间或空间区间内发生的事件数，且事件是随机、独立并以恒定平均速率λ发生，则X服从参数为λ（λ>0）的泊松分布。

We write X ~ Po(λ). The parameter λ is both the expected number of events and the variance of the distribution. Typical applications include the number of phone calls per minute, the number of typing errors per page, or the number of radioactive decays per second.

我们写作X ~ Po(λ)。参数λ既是事件的期望个数，也是该分布的方差。典型的应用包括每分钟电话呼入数、每页打字错误数或每秒放射性衰变数等。

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

For X ~ Po(λ), the probability of observing exactly x events is given by the probability mass function:

对于X ~ Po(λ)，恰好观察到x个事件的概率由概率质量函数给出：

P(X = x) = e⁻λ λ^x / x! , for x = 0, 1, 2, …

Here e ≈ 2.71828, λ is the mean rate, and x! ula is the foundation for all Poisson probability calculations, including cumulative ones.

这里e ≈ 2.71828，λ是平均速率，x!表示x的阶乘。该公式是所有泊松概率计算的基础，包括累积概率。

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3. Introducing the Cumulative Distribution Function (CDF) | 累积分布函数引入

The cumulative distribution function of a Poisson random variable is defined as the probability that X takes a value less than or equal to x:

泊松随机变量的累积分布函数定义为X取值小于或等于x的概率：

F(x) = P(X ≤ x) = Σ (from k = 0 to x) e⁻λ λ^k / k!

In other words, we simply sum the individual probabilities P(X = k) for k = 0, 1, 2, … fied to a closed form, so we rely on hand calculation, statistical tables, or calculators.

换句话说，我们只需将k = 0, 1, 2, …, x对应的各个概率P(X = k)相加。这个求和无法化简为封闭形式，因此我们依赖于手工计算、统计表格或计算器。

Because the Poisson CDF is discrete and non‑decreasing, it is particularly useful for answering questions like “What is the probability of no more than…” or “What is the probability of at least…?”.

由于泊松CDF是离散且非递减的，它在回答诸如“不超过…”或“至少…”的概率问题时特别实用。

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4. Calculating Poisson Cumulative Probabilities: Formula, Tables, and Calculators | 计算累积概率：公式、表格与计算器

sum term by term using the PMF. For instance, if λ = 2 and we need P(X ≤ 3), we calculate:

如果要手动计算P(X ≤ x)，可以使用PMF逐项累加。例如，当λ=2且需计算P(X ≤ 3)时，我们计算：

P(X=0) + P(X=1) + P(X=2) + P(X=3)

rid of λ and x values. You locate the row for your λ, read across to the column for x, and read the cumulative probability directly.

然而，若λ较大，这一过程会变得繁琐。Edexcel考试通常会提供泊松累积分布表的片段，表中列出了不同λ和x组合下的P(X ≤ x)。你只需找到对应λ的行，横读至x列，即可直接读取累积概率。

Modern calculators (such as the Casio ClassWiz series) also have built‑in Poisson CDF functions. By entering λ, lower bound 0, and upper bound x, the calculator returns the cumulative probability instantly. Being fluent with both tables and calculator use is essential for the exam.

现代计算器（如卡西欧ClassWiz系列）也内置了泊松CDF功能。输入λ、下限0和上限x，计算器就能立即返回累积概率。熟练掌握表格和计算器的使用对考试至关重要。

Below is a short extract from a typical cumulative Poisson table for λ = 2:

下面是一个典型的λ=2的泊松累积分布表的一部分：

x	P(X ≤ x)
0	0.1353
1	0.4060
2	0.6767
3	0.8571
4	0.9473

f you needed P(X = 3) you could subtract P(X ≤ 2) from P(X ≤ 3): 0.8571 − 0.6767 = 0.1804.

请注意，P(X ≤ 3) = 0.8571可从累积表中直接获得。如果需要P(X = 3)，你可以用P(X ≤ 3)减去P(X ≤ 2)：0.8571 − 0.6767 = 0.1804。

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5. Mean and Variance of the Poisson Distribution | 泊松分布的均值与方差

One of the charming features of the Poisson distribution is that both the mean and the variance equal λ. That is, for X ~ Po(λ):

泊松分布的一个引人注目的特点是，其均值和方差都等于λ。即，对于X ~ Po(λ)：

E(X) = Var(X) = λ

This relationship helps us check whether a data set might be modelled by a Poisson distribution: if the sample mean and sample variance are very different, a Poisson model is unlikely to be appropriate.

这一关系有助于我们检验一组数据是否可以用泊松分布建模：如果样本均值与样本方差差异很大，那么泊松模型很可能不适合。

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6. Finding ‘Greater Than’ and ‘Between’ Probabilities | 求解“大于”和“之间”的概率

the cumulative table typically provides P(X ≤ x), we use complement and difference rules:

在许多考题中，你会被要求求P(X > a)或P(a ≤ X ≤ b)。由于累积表通常提供P(X ≤ x)，我们利用补集和差分规则：

P(X > a) = 1 − P(X ≤ a)

P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1)

Poisson distribution is discrete, P(X ≥ a) = 1 − P(X ≤ a − 1) and P(X < a) = P(X ≤ a − 1).

注意严格不等号：由于泊松分布是离散的，有P(X ≥ a) = 1 − P(X ≤ a − 1)，以及P(X < a) = P(X ≤ a − 1)。

y that more than 3 calls arrive, we compute P(X > 3) = 1 − P(X ≤ 3) = 1 − 0.8571 = 0.1429.

例如，若λ=2，我们希望计算超过3通电话的概率，则P(X > 3) = 1 − P(X ≤ 3) = 1 − 0.8571 = 0.1429。

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7. Conditions for Using the Poisson Model | 泊松模型使用条件

Before applying the Poisson distribution, you must ensure that the context meets these criteria:

在应用泊松分布之前，必须确保情境满足以下条件：

• Events occur singly in time or space – two events cannot happen at exactly the same instant or point.
• 事件在时间或空间上单独发生——两个事件不能在同一瞬间或同一点发生。
• Events are independent – the occurrence of one event does not affect the probability of another.
• 事件相互独立——一个事件的发生不影响另一个事件发生的概率。
• The average rate λ is constant – the expected number of events per unit interval remains the same.
• 平均速率λ恒定——单位区间内期望的事件数保持不变。
• The interval can be divided into infinitely small sub‑intervals, and the probability of more than one event in a tiny sub‑interval is negligible.
• 区间可以无限细分，且在极小的子区间内发生多于一个事件的概率可忽略不计。

Failing to check these conditions often leads to an invalid model. In Edexcel questions you may be asked to comment on the suitability of a Poisson model.

未能检验这些条件往往会导致模型无效。在Edexcel试题中，你可能需要对泊松模型的适用性作出评价。

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8. Poisson Approximation to the Binomial Distribution | 泊松分布对二项分布的近似

This is extremely useful because binomial CDF calculations become cumbersome for large n.

当n很大且p很小时，二项分布B(n, p)可用参数λ = np的泊松分布来近似。这一方法极为有用，因为对于较大的n，二项CDF的计算会变得繁琐。

The rule of thumb is: n > 50 and np < 5, or n > 100 and np < 10. Under these conditions the approximation is very accurate. For the cumulative probability, we then use the Poisson CDF:

经验法则是：n > 50且np < 5，或n > 100且np < 10。在这些条件下近似非常精确。对于累积概率，我们随后使用泊松CDF：

P(X ≤ x) ≈ Σ (from k=0 to x) e^−np (np)^k / k!

1). Approximating using Poisson with λ = 200 × 0.01 = 2, we can quickly obtain P(X ≤ 3) ≈ 0.8571 from the table earlier.

假设一家工厂生产200件产品，每件次品概率为0.01。记X为次品数。准确分布为B(200,0.01)。用λ = 200 × 0.01 = 2的泊松近似，我们可从上表中快速得到P(X ≤ 3) ≈ 0.8571。

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9. Worked Examples with CDF | 累积分布函数例题

Example 1: A receptionist receives an average of 4 calls per hour. Find the probability that in a given hour she receives at most 2 calls.

例1：某接线员平均每小时收到4通电话。求在一给定小时内她最多接到2通电话的概率。

X ~ (X=0) + P(X=1) + P(X=2). Using a table or calculator yields P(X ≤ 2) = 0.2381 (to 4 d.p.).

X ~ Po(4)。P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2)。使用表格或计算器得 P(X ≤ 2) = 0.2381（保留4位小数）。

Example 2: A website experiences an average of 0.5 crashes per day. What is the probability that there are fewer than 3 crashes in a 5‑day week?

例2：某网站平均每天发生0.5次崩溃。问在5天的工作周内，崩溃次数少于3次的概率是多少？

Over 5 days λ = 5 × 0.5 = 2.5. Let Y ~ Po(2.5). P(Y < 3) = P(Y ≤ 2) = 0.5438 (using cumulative table or calculator).

在5天内，λ = 5 × 0.5 = 2.5。令Y ~ Po(2.5)。P(Y < 3) = P(Y ≤ 2) = 0.5438（使用累积表或计算器）。

Example 3 (approximation): ed 120 times. Let X be the number of sixes. Use a Poisson approximation to find P(X ≤ 15).

例3（近似）：一颗公平的骰子被掷120次。记X为出现6点的次数。用泊松近似求P(X ≤ 15)。

=20, we get P(X ≤ 15) ≈ 0.1565. For comparison, the exact binomial calculation would be very heavy.

此处n=120, p=1/6，故λ=np=20。尽管np>10，但λ=20的泊松近似依然可行。用λ=20查累积表或计算器得P(X ≤ 15) ≈ 0.1565。作为对比，精确的二项计算将非常繁琐。

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10. Common Mistakes and How to Avoid Them | 常见错误及避坑指南

A frequent error is misinterpreting inequalities. Always convert “at least” and “more than” carefully when using the cumulative table. Write down the complement explicitly before reaching for numbers.

一个常见错误是误判不等号。在使用累积表时，始终要小心转换“至少”和“大于”。在代入数字之前，明确写下补事件表达式。

Another pitfall is forgetting that the Poisson CDF is not continuous; therefore, no continuity correction is needed – confusing this with normal approximations is a classic exam mistake.

另一个陷阱是忘记泊松CDF并非连续，因此不需要连续性校正——将其与正态近似混淆是典型的考试错误。

Students sometimes sum the probabilities incorrectly when doing manual work. Always double‑check factorial calculations and e⁻λ values. Using a calculator’s CDF function eliminates most arithmetic errors.

学生在手工计算时有时会错误累加概率。务必核对阶乘计算和e⁻λ的值。使用计算器的CDF功能可消除大部分算术错误。

Finally, ensure that λ is adjusted when the interval length changes. For cumulative probabilities over a different length, update λ proportionally before consulting the table.

最后，确保在区间长度改变时调整λ。在查表前，若求不同长度上的累积概率，应按比例更新λ。

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11. Summary | 总结

The Poisson cumulative distribution function is a powerful tool for summing the probabilities of up to a certain number of rare events. By understanding both the underlying formula and the practical use of tables and calculators, you can confidently tackle Edexcel exam questions involving “less than”, “at most”, “more than”, or “between” probabilities. Remember the key conditions for using Poisson models, and use the Poisson approximation to the binomial when n is large and p is small to simplify complex problems. With practice, you will find the Poisson CDF both elegant and efficient.

泊松累积分布函数是一个强大的工具，用于累加至某一数量的稀有事件的概率。通过理解其底层公式以及表格和计算器的实际用法，你可以自信地应对Edexcel考试中涉及“小于”、“至多”、“多于”或“介于”的概率问题。记住使用泊松模型的关键条件，并当n大p小时用泊松近似二项以简化复杂问题。通过练习，你会发现泊松CDF既优雅又高效。

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