Mean and SD of Random Variables · 随机变量的均值与标准差
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| expected value/ekˈspektɪd ˈvæljuː/ | 期望值 | qī wàng zhí |
| variance/ˈveərɪəns/ | 方差 | fāng chà |
The long-run average
- The mean (or expected value 期望值) of a random variable is its long-run average outcome.
- Written $\mu_X$ or $E(X)$ — the balance point of the distribution.
- It's a weighted average: each value weighted by its probability.
- Over many repetitions, the average result closes in on $\mu_X$.
长期平均
- 随机变量的均值（或期望值）是它的长期平均结果。
- 记作 $\mu_X$ 或 $E(X)$——分布的平衡点。
- 它是一个加权平均：每个取值按它的概率加权。
- 在多次重复后，平均结果会逼近 $\mu_X$。
Computing the mean
-
$$\mu_X = \sum x_i \, P(x_i)$$
- Multiply each value by its probability, then add them all up.
- The mean need not be a possible value (a "family of $2.5$ kids").
- It's an average, not a prediction of any single trial.
计算均值
-
$$\mu_X = \sum x_i \, P(x_i)$$
- 把每个取值乘以它的概率，再全部加起来。
- 均值不必是一个可能的取值（“$2.5$ 个孩子的家庭”）。
- 它是一个平均，而非对任何单次试验的预测。
Variance and standard deviation
- The variance 方差 $\sigma_X^2 = \sum (x_i - \mu_X)^2 \, P(x_i)$ — the probability-weighted average squared distance from the mean.
- The standard deviation $\sigma_X = \sqrt{\sigma_X^2}$ brings it back to the original units.
- Bigger $\sigma_X$ = outcomes are more spread out around the mean.
- Square the deviations, weight by probability, sum, then square-root.
方差与标准差
- 方差 $\sigma_X^2 = \sum (x_i - \mu_X)^2 \, P(x_i)$——按概率加权的、离均值的平均平方距离。
- 标准差 $\sigma_X = \sqrt{\sigma_X^2}$ 把它带回原来的单位。
- $\sigma_X$ 越大 = 结果在均值周围越分散。
- 把偏差平方，按概率加权，求和，再开平方根。
Interpreting the SD
- $\sigma_X$ is the typical distance of an outcome from the mean $\mu_X$.
- Small $\sigma_X$: results cluster tightly near the expected value.
- Large $\sigma_X$: results swing widely from trial to trial.
- Together, $\mu_X$ (center) and $\sigma_X$ (spread) summarize the whole distribution.
解读标准差
- $\sigma_X$ 是一个结果离均值 $\mu_X$ 的典型距离。
- $\sigma_X$ 小：结果紧密聚集在期望值附近。
- $\sigma_X$ 大：结果在各次试验间大幅摆动。
- $\mu_X$（中心）和 $\sigma_X$（分散）一起概括了整个分布。
The expected value is a long-run average, not a guaranteed or even possible outcome. $E(X)=2.5$ children means the average over many families, not that any family has $2.5$ kids. And compute the SD from variance: square the deviations first, weight, sum, then take the square root — don't average the raw distances.
期望值是长期平均，而非有保证的、甚至可能的结果。$E(X)=2.5$ 个孩子指的是许多家庭的平均，而非任何一个家庭有 $2.5$ 个孩子。并且要从方差算标准差：先把偏差平方、加权、求和，然后再开平方根——不要直接平均原始距离。
A game: win $5$ dollars with probability $0.2$, else win $0$. Let $X$ be the winnings.
- Mean: $\mu_X = 5(0.2) + 0(0.8) = 1$ dollar — the long-run average payout.
- Variance: $(5-1)^2(0.2) + (0-1)^2(0.8) = 3.2 + 0.8 = 4$.
- SD: $\sigma_X = \sqrt{4} = 2$ dollars — typical distance from the $1$ average.
一个游戏：以概率 $0.2$ 赢 $5$ 元，否则赢 $0$ 元。设 $X$ 为赢得的钱（单位：元）。
- 均值：$\mu_X = 5(0.2) + 0(0.8) = 1$ 元——长期平均支付。
- 方差：$(5-1)^2(0.2) + (0-1)^2(0.8) = 3.2 + 0.8 = 4$。
- 标准差：$\sigma_X = \sqrt{4} = 2$ 元——离 $1$ 元平均的典型距离。
The mean (expected value) $\mu_X = \sum x_i P(x_i)$ is the long-run average — a probability-weighted average that needn't be a possible value. The variance $\sigma_X^2 = \sum (x_i-\mu_X)^2 P(x_i)$ and standard deviation $\sigma_X = \sqrt{\sigma_X^2}$ measure the typical distance of an outcome from $\mu_X$.
均值（期望值） $\mu_X = \sum x_i P(x_i)$ 是长期平均——一个概率加权平均，不必是可能的取值。方差 $\sigma_X^2 = \sum (x_i-\mu_X)^2 P(x_i)$ 和标准差 $\sigma_X = \sqrt{\sigma_X^2}$ 衡量一个结果离 $\mu_X$ 的典型距离。
Weighting values by probability · 按概率给取值加权
The mean is the balance point of this weighted distribution. · 均值是这个加权分布的平衡点。
X wins 5 with prob 0.2, else 0. Find the mean E(X) = 5(0.2)+0(0.8). · X 以概率 0.2 赢 5，否则 0。求均值 E(X) = 5(0.2)+0(0.8)。
5(0.2) + 0(0.8) = 1. · 5(0.2) + 0(0.8) = 1。
With mean 1: variance = (5−1)²(0.2)+(0−1)²(0.8). Then find the standard deviation. · 均值为 1：方差 = (5−1)²(0.2)+(0−1)²(0.8)。再求标准差。
Variance = 3.2 + 0.8 = 4, so SD = √4 = 2. · 方差 = 3.2 + 0.8 = 4，所以标准差 = √4 = 2。
The expected value of a random variable must be one of its possible values. · 随机变量的期望值必须是它的某个可能取值。
It's a long-run average — e.g. 2.5 children — need not be attainable. · 它是长期平均——如 2.5 个孩子——不必是能达到的值。
The standard deviation of a random variable measures... · 随机变量的标准差衡量……
SD = typical spread around the mean. · 标准差 = 均值周围的典型分散。
The mean of a random variable is also called its ___ value (one word). · 随机变量的均值也叫它的 ___ 值（填英文一词 expected）。
Mean = expected value = long-run average. · 均值 = 期望值 = 长期平均。