Binomial Parameters · 二项分布的参数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| parameters/pəˈræmɪtəz/ | 参数 | cān shù |
The binomial mean
- For a binomial variable, the mean has a beautifully simple formula:
-
$$\mu_X = np$$
- Trials times success-probability — the expected number of successes.
- Flip a coin $100$ times: expect $\mu_X = 100 \times 0.5 = 50$ heads.
二项均值
- 对二项变量，均值有一个漂亮而简单的公式：
-
$$\mu_X = np$$
- 试验次数乘以成功概率——期望的成功次数。
- 抛硬币 $100$ 次：期望 $\mu_X = 100 \times 0.5 = 50$ 个正面。
The binomial standard deviation
- The standard deviation is:
-
$$\sigma_X = \sqrt{np(1-p)}$$
- It measures how much the count of successes typically varies.
- These two parameters 参数 ($\mu_X$, $\sigma_X$) summarize the whole binomial distribution.
二项标准差
- 标准差是：
-
$$\sigma_X = \sqrt{np(1-p)}$$
- 它衡量成功计数通常变动多少。
- 这两个参数（$\mu_X$、$\sigma_X$）概括了整个二项分布。
The shape
- The shape depends on $p$: at $p = 0.5$ the distribution is symmetric.
- $p < 0.5$ → skewed right; $p > 0.5$ → skewed left.
- As $n$ grows, the shape becomes more symmetric and bell-like (even for skewed $p$).
- This near-normal shape for large $n$ powers the inference in later units.
形状
- 形状取决于 $p$：在 $p = 0.5$ 时分布是对称的。
- $p < 0.5$ → 右偏；$p > 0.5$ → 左偏。
- 随着 $n$ 增大，形状变得更对称、更像钟形（即使 $p$ 偏斜）。
- 大 $n$ 时这种接近正态的形状，驱动了后面单元的推断。
Interpret in context
- Always state the parameters in the problem's own words.
- $\mu_X$: "we expect about $np$ successes on average."
- $\sigma_X$: "the number of successes typically varies by about $\sigma_X$."
- Numbers mean little until tied back to the real situation.
结合语境解读
- 总要用题目自己的话陈述参数。
- $\mu_X$：“我们平均期望大约 $np$ 次成功。”
- $\sigma_X$：“成功次数通常变动大约 $\sigma_X$。”
- 数字在与真实情境挂钩之前意义不大。
The tidy formulas $\mu=np$ and $\sigma=\sqrt{np(1-p)}$ are for the binomial setting only — don't use them for a general random variable, and only after the BINS conditions check out. And note the SD uses $p(1-p)$: it's largest at $p=0.5$ and shrinks as $p$ nears $0$ or $1$ (extreme $p$ makes the count more predictable).
整洁的公式 $\mu=np$ 和 $\sigma=\sqrt{np(1-p)}$ 仅适用于二项设定——不要把它们用于一般随机变量，且只在 BINS 条件通过之后才用。并且注意标准差用了 $p(1-p)$：它在 $p=0.5$ 时最大，在 $p$ 接近 $0$ 或 $1$ 时缩小（极端的 $p$ 让计数更可预测）。
A free-throw shooter makes $p = 0.8$; she takes $n = 25$ shots. $X =$ makes.
- Mean: $\mu_X = 25 \times 0.8 = 20$ makes expected.
- SD: $\sigma_X = \sqrt{25 \times 0.8 \times 0.2} = \sqrt{4} = 2$.
- Shape: $p = 0.8 > 0.5$ → skewed left; with $n=25$ it's fairly bell-shaped.
一名罚球手命中率 $p = 0.8$；她投 $n = 25$ 次。$X =$ 命中数。
- 均值：$\mu_X = 25 \times 0.8 = 20$ 次期望命中。
- 标准差：$\sigma_X = \sqrt{25 \times 0.8 \times 0.2} = \sqrt{4} = 2$。
- 形状：$p = 0.8 > 0.5$ → 左偏；$n=25$ 时相当接近钟形。
For a binomial variable, the parameters are $\mu_X = np$ and $\sigma_X = \sqrt{np(1-p)}$. The shape is symmetric at $p=0.5$, skewed otherwise, and more bell-like as $n$ grows. Always interpret $\mu_X$ and $\sigma_X$ in context — expected successes and their typical variation.
对二项变量，参数为 $\mu_X = np$ 和 $\sigma_X = \sqrt{np(1-p)}$。形状在 $p=0.5$ 时对称，否则偏斜，且随 $n$ 增大而更像钟形。总要结合语境解读 $\mu_X$ 和 $\sigma_X$——期望成功数及其典型变动。
Shape of a binomial distribution · 二项分布的形状
With p ≠ 0.5 the distribution is skewed; larger n looks more bell-like. · 当 p ≠ 0.5 时分布偏斜；n 越大越像钟形。
A binomial has n = 25, p = 0.8. Find the mean np. · 一个二项分布 n = 25、p = 0.8。求均值 np。
μ = np = 25 × 0.8 = 20. · μ = np = 25 × 0.8 = 20。
For n = 25, p = 0.8, find the standard deviation √(np(1−p)). · 对 n = 25、p = 0.8，求标准差 √(np(1−p))。
√(25·0.8·0.2) = √4 = 2. · √(25·0.8·0.2) = √4 = 2。
A binomial distribution is exactly symmetric when... · 二项分布恰好对称是在……
p = 0.5 gives a symmetric distribution. · p = 0.5 给出对称分布。
As n increases, a binomial distribution becomes more bell-shaped (more symmetric). · 随着 n 增大，二项分布变得更像钟形（更对称）。
Large n pushes the shape toward normal. · 大 n 把形状推向正态。
The binomial standard deviation √(np(1−p)) is largest when... · 二项标准差 √(np(1−p)) 在……时最大。
p(1−p) peaks at p = 0.5. · p(1−p) 在 p = 0.5 处最大。