The Geometric Distribution · 几何分布
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| geometric/ˌdʒiːəʊˈmetrɪk/ | 几何 | jǐ hé |
| geometric random variable/ˌdʒiːəʊˈmetrɪk ˈrændəm ˈveərɪəbl/ | 几何随机变量 | jǐ hé suí jī biàn liàng |
Waiting for the first success
- The geometric 几何 distribution counts trials until the first success.
- A geometric random variable 几何随机变量 $X$ = the number of trials to get the first success.
- Example: how many rolls until you get your first six?
- Unlike binomial, $n$ isn't fixed — you stop the moment you succeed.
等待第一次成功
- 几何分布数的是到第一次成功为止的试验次数。
- 几何随机变量 $X$ = 得到第一次成功所需的试验次数。
- 例如：掷多少次才第一次掷出六点？
- 与二项不同，$n$ 不固定——你一旦成功就停下。
The geometric setting
- Same trial conditions as binomial, but a different stopping rule:
- Repeated independent trials, each with the same probability $p$ of success.
- You keep going until the first success, then stop.
- "Until first success" is the signal word for geometric.
几何设定
- 试验条件与二项相同，但停止规则不同：
- 重复的独立试验，每次成功的概率相同为 $p$。
- 你一直进行直到第一次成功，然后停止。
- “直到第一次成功”是几何分布的信号词。
The geometric formula
-
$$P(X = k) = (1-p)^{k-1}\,p$$
- To succeed first on trial $k$: fail the first $k-1$ times, then succeed.
- $(1-p)^{k-1}$ = those failures; $p$ = the success on trial $k$.
- The probabilities shrink as $k$ grows — early success is most likely.
几何公式
-
$$P(X = k) = (1-p)^{k-1}\,p$$
- 要在第 $k$ 次首次成功：前 $k-1$ 次失败，然后成功。
- $(1-p)^{k-1}$ = 那些失败；$p$ = 第 $k$ 次的成功。
- 概率随 $k$ 增大而缩小——早成功最有可能。
The geometric mean
- The mean (expected number of trials) is:
-
$$\mu_X = \frac{1}{p}$$
- If $p = 0.2$, expect $1/0.2 = 5$ trials on average until the first success.
- Rarer successes ($small$p$) mean you wait longer — makes intuitive sense.
几何均值
- 均值（期望的试验次数）是：
-
$$\mu_X = \frac{1}{p}$$
- 若 $p = 0.2$，平均期望 $1/0.2 = 5$ 次试验才第一次成功。
- 成功越罕见（$p$ 小）等得越久——符合直觉。
Geometric vs. binomial comes down to the stopping rule. Binomial: fixed $n$, count successes. Geometric: run until the first success, count trials. If a problem fixes the number of tries, it's binomial; if it says "until the first…", it's geometric. And the geometric mean is $1/p$ — not $np$.
几何与二项的区别在于停止规则。二项：固定 $n$，数成功数。几何：进行到第一次成功，数试验次数。若题目固定了尝试次数，就是二项；若说“直到第一次……”，就是几何。而且几何均值是 $1/p$——不是 $np$。
Roll a die until the first six; $p = 1/6$.
- $P(X = 3) = (5/6)^2 (1/6) = \tfrac{25}{36}\cdot\tfrac{1}{6} \approx 0.116$ — first six on the $3$rd roll.
- Mean: $\mu_X = 1/(1/6) = 6$ rolls expected until the first six.
- Makes sense — a six comes up on average once every six rolls.
掷骰子直到第一次出现六点；$p = 1/6$。
- $P(X = 3) = (5/6)^2 (1/6) = \tfrac{25}{36}\cdot\tfrac{1}{6} \approx 0.116$——第 $3$ 次掷出首个六点。
- 均值：$\mu_X = 1/(1/6) = 6$ 次——期望掷 $6$ 次才第一次出六点。
- 有道理——六点平均每六次出现一次。
A geometric random variable counts trials until the first success, with independent trials at constant $p$. Then $P(X=k)=(1-p)^{k-1}p$ and the mean is $\mu_X = \frac{1}{p}$. The key contrast with binomial: geometric runs until the first success (variable trials), binomial has a fixed $n$.
几何随机变量数的是到第一次成功为止的试验次数，试验独立、$p$ 恒定。此时 $P(X=k)=(1-p)^{k-1}p$，均值为 $\mu_X = \frac{1}{p}$。与二项的关键区别：几何进行到第一次成功（试验次数可变），二项有固定的 $n$。
Rolling until the first success · 掷到第一次成功为止
Keep rolling until the target appears — that's a geometric count. · 一直掷到目标出现——这就是一个几何计数。
A geometric variable has p = 0.2. Find the mean number of trials, 1/p. · 一个几何变量 p = 0.2。求期望试验次数 1/p。
μ = 1/p = 1/0.2 = 5. · μ = 1/p = 1/0.2 = 5。
Which situation is geometric (not binomial)? · 哪种情形是几何（而非二项）？
'Until the first success' → geometric; fixed n → binomial. · “直到第一次成功” → 几何；固定 n → 二项。
Roll a die until the first six (p = 1/6). Find the expected number of rolls. · 掷骰子直到第一次六点（p = 1/6）。求期望的投掷次数。
μ = 1/p = 1/(1/6) = 6. · μ = 1/p = 1/(1/6) = 6。
In the geometric formula P(X=k) = (1−p)^(k−1)·p, the (1−p)^(k−1) part is the first k−1 failures. · 在几何公式 P(X=k) = (1−p)^(k−1)·p 中，(1−p)^(k−1) 部分是前 k−1 次失败。
Fail k−1 times, then succeed on trial k. · 先失败 k−1 次，再在第 k 次成功。
The mean of a geometric random variable is 1 divided by ___ (the letter for success probability). · 几何随机变量的均值是 1 除以 ___（成功概率的字母）。
μ = 1/p. · μ = 1/p。