📚 2B Negative Binomial Distribution | 2B 负二项分布
In probability theory, the negative binomial distribution extends the geometric distribution by modelling the number of independent Bernoulli trials needed to achieve a fixed number of successes. Where the binomial distribution fixes the number of trials and counts successes, the negative binomial fixes the number of successes and counts the trials, which makes it the natural tool for waiting-time and "how long until" questions.
在概率论中，负二项分布是几何分布的推广，它描述了在一系列独立的伯努利试验中，获得固定次数成功所需的试验次数。二项分布固定试验次数、统计成功次数；负二项分布则固定成功次数、统计所需试验次数，因此它天然适用于等待时间与"还要多久才能完成"这一类问题。
1. The Waiting-Time Idea | 等待时间的思想
The binomial model answers a fixed-length question: run n trials, then ask how many successes appeared. The negative binomial model answers an open-ended question: keep running trials until r successes have appeared, then ask how many trials that took. The random quantity is therefore the time (or number of attempts) rather than the count of successes.
二项模型回答的是一个长度固定的问题：做 n 次试验，然后问其中有多少次成功。负二项模型回答的是一个开放式的问题：一直做试验直到出现 r 次成功，然后问这共花掉了多少次试验。因此随机变量是"时间"（即试验次数），而不是成功次数。
Typical real-world settings in which this model appears include:
该模型常见的实际应用场景包括：
- A basketball player keeps shooting until 3 baskets are scored; how many attempts are needed?
一名篮球运动员持续投篮，直到命中 3 球为止，需要投多少次？ - A quality inspector examines items until 5 conforming items have been found.
质检员一件一件检查产品，直到找到 5 件合格品为止。 - A call centre keeps dialling until 4 customers answer the phone.
呼叫中心持续拨号，直到有 4 位客户接听电话为止。 - A gambler rolls a die until a 6 has appeared 4 times.
赌客反复掷骰子，直到 6 点出现 4 次为止。
ability p. Setting r = 1 reduces the whole model to the geometric distribution, which is the simplest waiting-time distribution of all.
在所有这些例子中，试验次数是随机的，成功次数是事先固定的，各次试验相互独立且成功概率恒为 p。令 r = 1，整个模型就退化为几何分布，即最简单的等待时间分布。
2. Definition and Model Setup | 定义与模型设定
ity 1 − p. The trials continue until the r-th success is observed, where r is a fixed positive integer. It is convenient to write q = 1 − p for the failure probability.
负二项实验由重复的独立试验组成，每次试验以概率 p 获得成功，以概率 1 − p 失败。试验持续进行，直到观察到第 r 次成功为止，其中 r 是固定的正整数。为方便书写，记 q = 1 − p 为失败概率。
The defining assumptions can be listed explicitly, because exam questions often test whether a situation really fits the model:
定义中的假设可以逐条列出，因为考题常常考查某个情境是否真的符合该模型：
- 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 p is the same on every trial.
每次试验的成功概率 p 都相同。 - The trials are mutually independent.
各次试验相互独立。 - The number of successes r is fixed before the experiment starts, and the trials stop as soon as the r-th success occurs.
成功次数 r 在实验开始前就已固定，一旦第 r 次成功出现，试验立即停止。
quired to obtain r successes. Hence X takes values x = r, r + 1, r + 2, ... and we write X ~ th success occurs, so X is always at least r, and X can be arbitrarily large in principle.
我们定义随机变量 X 为获得 r 次成功所需的试验次数。因此 X 的取值为 x = r, r + 1, r + 2, ...，记作 X ~ NB(r, p)。等价地说，X 就是第 r 次成功发生的那一次试验的序号，所以 X 恒不小于 r，而理论上 X 可以任意大。
3. Probability Mass Function | 概率质量函数
If X ~ lity mass function (PMF) is given by the following formula, valid for every integer x ≥ r.
若 X ~ NB(r, p) 表示获得 r 次成功所需的试验次数，则其概率质量函数 (PMF) 由下式给出，对一切整数 x ≥ r 成立。
P(X = x) = C(x − 1, r − 1) × pʳ × (1 −
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