📚 Poisson Cumulative Distribution Function | 泊松累积分布函数
It gives the probability that a Poisson random variable takes a value less than or equal to a chosen number. Unlike the probability mass function, which returns P(X = x) for one exact count, the cumulative distribution function adds probabilities from x = 0 upward. This article explains how to calculate, interpret and apply the Poisson CDF, including table use, calculator methods, complement rules, interval probabilities and the binomial approximation.
泊松累积分布函数是 Edexcel A-Level 数学的核心考点。它给出泊松随机变量取值小于或等于某个给定数值的概率。与给出单个取值概率 P(X = x) 的概率质量函数不同，累积分布函数从 x = 0 开始逐项累加概率。本文讲解泊松累积分布函数的计算、解释与应用，包括查表方法、计算器方法、补事件规则、区间概率以及二项分布的泊松近似。
1. What is the Poisson Cumulative Distribution Function? | 什么是泊松累积分布函数？
The cumulative distribution function of a Poisson random variable X ~ rom 0 up to k. For example, P(X ≤ 3) means the probability that the count is 0, 1, 2 or 3.
泊松随机变量 X ~ Po(λ) 的累积分布函数记为 F(k) = P(X ≤ k)。它把从 0 到 k 的每一个可能结果的概率累加起来。例如 P(X ≤ 3) 表示计数为 0、1、2 或 3 的总概率。
ct value, while the cumulative distribution function gives a running total. The CDF is especially useful for interval probabilities and for finding upper-tail probabilities such as P(X > k), because it turns a long sum into a single table or calculator value.
概率质量函数给出 P(X = x)，即恰好取某一个值的概率；而累积分布函数给出的是累加和。它在求区间概率以及求上尾概率（如 P(X > k)）时特别有用，因为它把一个很长的求和式压缩成一个表值或计算器结果。
F(k; λ) = P(X ≤ k) = Σ P(X = r), r = 0, 1, 2, ..., k
2. Notation, Domain and Key Properties | 记号、取值范围与关键性质
not be negative or fractional. The domain is therefore k = 0, 1, 2, 3, ... and the cumulative probability is constant between consecutive integers.
泊松累积分布函数只在非负整数 k 上有定义，因为泊松变量统计的是一个固定区间内事件发生的次数，不可能为负，也不能取分数。因此定义域为 k = 0, 1, 2, 3, ...，在相邻整数之间累积概率保持不变。
not be negative. Second, F(k) approaches 1 as k becomes large, since the total probability must be 1. Third, F(0) = P(X = 0) = e⁻λ, which is the starting value of the sum. Fourth, it is standard to treat F(−1) as 0, which makes interval formulas work cleanly for the lower tail.
有四个性质值得记住。第一，F(k) 关于 k 单调不减，因为每一项都是概率，概率不会为负。第二，当 k 足够大时 F(k) 趋近于 1，因为总概率必须等于 1。第三，F(0) = P(X = 0) = e⁻λ，这是整个求和式的起点。第四，通常约定 F(−1) = 0，这样区间公式在下端也能统一书写。
ion is right-skewed, P(X ≤ λ) is usually slightly greater than one half. For example, if X ~ quick sanity check: an answer of 0.15 or 0.95 for P(X ≤ λ) is almost certainly a mistake.
由于泊松分布右偏，P(X ≤ λ) 通常略大于二分之一。例如当 X ~ Po(4.2) 时，P(X ≤ 4) ≈ 0.5898。这可以作为一个快速的合理性检查：如果 P(X ≤ λ) 算出 0.15 或 0.95，几乎可以肯定出错了。
3. The Cumulative Probability Formula | 累积概率公式
Substituting the Poisson probability mass function P(X = r) = e⁻λ λ^r / r! ility formula. The CDF is therefore a finite sum of an exponential factor and power terms.
将泊松概率质量函数 P(X = r) = e⁻λ λ^r / r! 代入求和式，就得到累积概率公式。因此累积分布函数是一个由指数因子和幂次项组成的有限和。
P(X ≤ k) = e⁻λ × Σ (λ^r / r!), r = 0, 1, 2, ..., k
convenient for hand calculation, especially when k is small: P(X ≤ k) = e⁻λ (1 + λ + λ²/2! + λ³/3! + ... + λ^k/k! Each bracket term is the individual Poisson probability with the e⁻λ factor taken outside.
把求和式展开可以得到便于手算的形式，尤其是 k 较小时：P(X ≤ k) = e⁻λ (1 + λ + λ²/2! + λ³/3! + ... + λ^k/k!)。括号内的每一项就是把 e⁻λ 提到外面之后的单个泊松概率。
There is no simple closed form for this sum, so numerical values are found by adding terms one at a time, by reading a cumulative Poisson table, or by using the distribution function on a calculator. For most exam questions you will use a calculator or a table after writing down the correct probability statement, and method marks are awarded for the statement itself.
这个求和式没有简单的闭式表达式，因此数值通常通过逐项相加、查泊松累积表或使用计算器的分布函数来得到。在大多数考试题中，你要先写出正确的概率陈述，再用计算器或表格求值，而方法分正是给在概率陈述这一步。
4. Reading Poisson Cumulative Tables | 如何使用泊松累积表
he number you read is P(X ≤ k), not P(X = k), so always check the heading before using it.
泊松累积表按参数 λ 和取值 k 排列。先找到你所用的 λ 所在的行或列，再对应到 k 的那一格，读出的数值就是 P(X ≤ k)，而不是 P(X = k)。使用前一定要先看清表头说明。
If your λ is not listed in the table, do not attempt linear interpolation between two columns, because the cumulative probabilities are not linear in λ. Use a calculator instead, or restate the problem using a λ that does appear in the table. Table values are usually given to four decimal places, so keep four decimal places in your working and rounding.
如果所用的 λ 不在表中，不要在两列之间做线性插值，因为累积概率关于 λ 并不是线性的。此时改用计算器，或者把题目改写成表内存在的 λ。表值通常给到四位小数，因此计算过程和最终答案都保留四位小数。
| k | P(X ≤ k) for Po(4.2) | P(X = k) for Po(4.2) |
| 0 | 0.0150 | 0.0150 |
| 1 | 0.0780 | 0.0630 |
| 2 | 0.2102 | 0.1323 |
| 3 | 0.3954 | 0.1852 |
| 4 | 0.5898 | 0.1944 |
| 5 | 0.7531 | 0.1633 |
| 6 | 0.8675 | 0.1143 |
s exactly the relationship that makes the cumulative table so powerful in reverse as well.
注意 P(X = k) 这一列正好是相邻两个累积值之差：P(X = 3) = F(3) − F(2) = 0.3954 − 0.2102 = 0.1852。正是这一关系使得累积表在反向上也同样有用。
5. Calculator Methods | 计算器方法
Modern graphical and scientific calculators provide a built-in Poisson cumulative function. On a TI-Nspire, the path is Menu, then 6 Statistics, then 5 Distributions, then E: Poisson Cdf. The syntax is poissCdf(λ, lower bound, upper bound), where the lower bound is 0 for a lower-tail probability.
现代图形计算器和科学计算器都内置泊松累积分布函数。在 TI-Nspire 上，路径是 Menu，再选 6 Statistics、5 Distributions、E: Poisson Cdf。语法为 poissCdf(λ, 下限, 上限)，求下尾概率时下限填 0。
, then the value of x. The calculator returns P(X ≤ x), which is exactly F(k). On a TI-84, use poissoncdf(λ, k) from the DISTR menu.
在 Casio fx-CG50 或 fx-991 Classwiz 上，按 MENU，选 7 Distribution，选 Poisson，再选 CD（累积分布）。先输入 λ，再输入 x 的值，计算器返回的就是 P(X ≤ x)，即 F(k)。在 TI-84 上，从 DISTR 菜单中选择 poissoncdf(λ, k)。
- Check which parameter comes first: most calculators ask for λ before x.
- 先确认参数顺序：多数计算器先输入 λ，再输入 x。
- Some models return P(X ≤ x) only for integer x, so round counts to integers first.
- 部分机型只在 x 为整数时返回 P(X ≤ x)，因此要先把计数取整。
- Always quote four decimal places unless the question asks for a different accuracy.
- 除题目另有精度要求，答案一律保留四位小数。
6. The Complement Rule and Upper-Tail Probabilities | 补事件规则与上尾概率
ower-tail values, the complement rule is essential: P(X > k) = 1 − P(X ≤ k) = 1 − F(k).
很多考题问的是大于某个值的概率。由于表格和计算器通常只给出下尾概率，补事件规则就非常关键：P(X > k) = 1 − P(X ≤ k) = 1 − F(k)。
= 1 − F(k − 1). Similarly, P(X < k) = F(k − 1), because the value k itself is excluded.
最容易出错的地方是区分大于和大于或等于。求 P(X ≥ k) 时要去掉所有小于 k 的值，但 k 本身包含在内，因此 P(X ≥ k) = 1 − F(k − 1)。同理 P(X < k) = F(k − 1)，因为 k 本身被排除。
gt; k) = 1 − F(k) P(X ≥ k) = 1 − F(k − 1) P(X < k) = F(k − 1)
When using a calculator, you can avoid the complement rule entirely by entering the lower and upper bounds directly, for example poissCdf(4.2, 7, 10⁹) for P(X > 6). Setting a very large upper bound is accepted in practice but writing the full probability statement first still earns method marks.
使用计算器时，可以直接输入上下限而完全避开补事件规则，例如用 poissCdf(4.2, 7, 10⁹) 求 P(X > 6)。把上限设得很大在实际上是可以接受的，但先写出完整的概率陈述仍然能拿到方法分。
7. Interval Probabilities | 区间概率
Interval probabilities are the main reason the CDF exists. To find the probability that X lies between two values, subtract the cumulative probability of everything below the interval from the cumulative probability of everything up to the top of the interval.
区间概率是累积分布函数存在的主要理由。要求 X 落在两个值之间的概率，只需用上限处的累积概率减去区间下方所有取值的累积概率。
P(a ≤ X ≤ b) = F(b) − F(a − 1)
case X = a, which is supposed to be included when the inequality is written as a ≤ X ≤ b.
这里减 a − 1 而不是减 a，是学生最容易出错的步骤。如果写成 F(a)，就会把 X = a 这个本应包含在 a ≤ X ≤ b 中的情况排除掉。
F(b) − F(a). Always sketch a small number line marking which endpoints are included, then translate that into the correct cumulative values.
如果是严格不等式，端点要做相应调整：P(a < X < b) = F(b − 1) − F(a)，P(a < X ≤ b) = F(b) − F(a)。先在草稿上画一条小数轴，标出哪些端点包含在内，再翻译成正确的
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图