林深见鹿（四）：概率论与数理统计（3）

摘要：Today, the editor brings the "Deep in the Woods, the Deer Appears (Part 4): Probability Theory and Mathematical Statistics (3)".

分享兴趣，传播快乐，增长见闻，留下美好！

亲爱的您，这里是LearningYard新学苑。

今天小编为大家带来

“林深见鹿（四）：概率论与数理统计（3）”。

欢迎您的访问！

Share interest, spread happiness, increase knowledge, and leave beautiful.

Dear, this is the LearingYard New Academy!

Today, the editor brings the "Deep in the Woods, the Deer Appears (Part 4): Probability Theory and Mathematical Statistics (3)".

Welcome to visit!

思维导图

MindMapping

翻开概率论习题册：“某机器生产零件次品率为5%，求100件中恰有3件次品的概率？”“某路口一天发生事故数服从λ=2的分布，求一天内至少发生1起事故的概率？”“抽卡出金概率为1%，求第50抽才第一次出金的概率？”你是否也曾困惑——这些题目不都是求概率吗？难道用的不是同一个公式？如果你正在为区分这几种常见的离散分布而头疼，那么这篇“鉴宝图鉴”正是为你准备的。让我们一同揭开二项分布、泊松分布和几何分布的神秘面纱。

Open the probability theory exercise book: "A machine produces parts with a defect rate of 5%. What is the probability of having exactly 3 defective parts out of 100?" "The number of accidents at a certain intersection per day follows a distribution with λ=2. What is the probability of at least one accident occurring in a day?" "The probability of drawing a golden card is 1%. What is the probability of drawing a golden card for the first time on the 50th draw?" Have you ever been confused—aren't all these questions asking for probabilities? Don't they all use the same formula? If you're struggling to distinguish between these common discrete distributions, then this "Treasure Identification Guide" is just for you. Let's uncover the mysteries of the Binomial distribution, Poisson distribution, and geometric distribution together.

二项分布：多重试验的计数专家

Binomial Distribution: The Counting Expert for Multiple Trials

二项分布源于重复多次的伯努利试验，堪称离散分布家族中的“多重钥匙”，能够开启多扇相同的门。它的核心特征包括：n次独立的试验，每次试验只有两种可能结果（成功/失败），且每次成功的概率p保持不变。当我们询问“在n次试验中，成功k次的概率是多少？”时，就是在使用二项分布。这个分布在现实生活中有着广泛的应用：学霸计算n次考试中恰好及格k次的概率；质检员统计n个零件中恰好有k个次品的概率；甚至游戏玩家计算n连抽恰好抽中k张SSR的概率，都属于二项分布的范畴。

The binomial distribution originates from repeated Bernoulli trials and can be regarded as the "multi-key" within the family of discrete distributions, capable of unlocking multiple identical doors. Its core characteristics include: n independent trials, where each trial has only two possible outcomes (success/failure), and the probability of success p remains constant for each trial. When we ask, "What is the probability of achieving success k times in n trials?" we are utilizing the binomial distribution. This distribution has a wide range of applications in real life: top students calculating the probability of passing exactly k out of n exams; quality inspectors tallying the probability of having exactly k defective parts among n; and even gamers computing the probability of drawing exactly k SSR cards in n consecutive draws—all fall within the realm of the binomial distribution.

泊松分布：稀有事件的记录者

Poisson Distribution: The Recorder of Rare Events

泊松分布如同一个神秘的陶罐，稀有事件会随机地从时间里“冒”出来。它专门描述稀有事件在固定时间/空间内发生次数的概率。其核心特征是：事件的发生是独立且随机的，已知单位时间内发生的平均次数λ，且事件相继发生的时间间隔非常短。当我们探究“在特定范围内，某件事发生了k次？”时，泊松分布就派上了用场。客服人员统计一天内接到电话的次数，交通管理员记录一个路口一天内发生交通事故的次数，甚至编辑检查一页书上出现错别字的个数，都可以使用泊松分布。

The Poisson distribution is like a mysterious clay jar from which rare events randomly "emerge" over time. It specifically describes the probability of the number of times a rare event occurs within a fixed time or space. Its core characteristics are that the occurrence of events is independent and random, with a known average number of times λ they occur per unit of time, and the time intervals between successive events are extremely short. When we investigate "How many times does a certain event occur within a specific range?", the Poisson distribution comes into play. Customer service representatives tallying the number of phone calls received in a day, traffic administrators recording the number of traffic accidents at an intersection in a day, and even editors checking the number of typographical errors on a page of a book can all utilize the Poisson distribution.

几何分布：首次成功的等待者

Geometric Distribution: The Waiter for the First Success

几何分布就像一把需要耐心解锁的宝剑，失败多次，只为一朝成功。它专门计算首次成功所需的试验次数。其特征包括：同样是独立的伯努利试验，每次成功概率p不变，但充满了“等待”的悲情色彩。当我们询问“我等了多久才第一次成功？”时，就是在使用几何分布。游戏玩家想知道第一次抽中SSR是在第几次抽卡；表白者想知道第一次被答应是在第几次表白；射手想知道第一次命中目标是在第几枪——这些都是几何分布的典型应用。

The geometric distribution resembles a precious sword that requires patience to unlock, enduring multiple failures in anticipation of a single triumph. It specifically calculates the number of trials needed to achieve the first success. Its characteristics include: independent Bernoulli trials with a constant probability of success p for each trial, but imbued with a sense of tragic anticipation in "waiting." When we ask, "How long did I wait until my first success?", we are employing the geometric distribution. Game players wondering on which draw they will first obtain an SSR card; admirers wanting to know on which attempt they will finally receive a positive response to their confession; marksmen eager to find out on which shot they will first hit the target—these are all classic applications of the geometric distribution.

三大分布对比与总结

Comparison and Summary of the Three Major Distributions

为了更好地理解和区分这三大分布，我们可以从核心问题、参数、期望、方差和适用场景五个方面进行对比。二项分布解决的是“n次中成功k次”的问题，参数为n（次数）和p（概率），期望为np，方差为np(1-p)，适用于有明确次数限制的重复试验。泊松分布解决的是“范围内发生k次”的问题，参数为λ（平均次数），期望和方差均为λ，适用于稀有事件在固定间隔内的发生数。几何分布解决的是“首次成功在第k次”的问题，参数为p（概率），期望为1/p，方差为(1-p)/p²，适用于等待第一次成功所需的时间。在使用这些分布时，只需三步：首先读题，找到题目的“灵魂发问”；然后对号入座，看它符合哪个分布的特点；最后套用公式，直接计算。

To better understand and differentiate these three key distributions, we can compare them across five aspects: core problem, parameters, expectation, variance, and applicable scenarios. The binomial distribution addresses the issue of "achieving success k times in n trials," with parameters n (number of trials) and p (probability of success), an expectation of np, and a variance of np(1-p). It is suitable for repeated trials with a clearly defined number of attempts. The Poisson distribution deals with the problem of "an event occurring k times within a certain range," characterized by the parameter λ (average number of occurrences), with both expectation and variance equal to λ. It is applicable to counting the number of rare events occurring within a fixed interval. The geometric distribution resolves the question of "the first success occurring on the k-th trial," with the parameter p (probability of success), an expectation of 1/p, and a variance of (1-p)/p². It is used for situations involving the waiting time until the first success. When applying these distributions, follow these three steps: first, read the question carefully to identify its "core inquiry"; next, match the problem to the characteristics of one of the distributions; finally, apply the appropriate formula to calculate directly.

今天的分享就到这里了，

如果您对文章有独特的想法，

欢迎给我们留言。

让我们相约明天，

祝您今天过得开心快乐！

That's all for today's sharing.

If you have a unique idea about the article,

please leave us a message,

and let us meet tomorrow.

I wish you a nice day!

翻译：文心一言

参考资料：百度百科