Combining Random Variables · 组合随机变量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| linear transformation/ˈlɪnɪə trænsfɔːˈmeɪʃn/ | 线性变换 | xiàn xìng biàn huàn |
Scaling and shifting
- A linear transformation 线性变换 $aX + b$ scales by $a$ and shifts by $b$.
- Mean: $\mu_{aX+b} = a\,\mu_X + b$ — both scale and shift carry through.
- SD: $\sigma_{aX+b} = |a|\,\sigma_X$ — scaling stretches spread; shifting doesn't.
- Adding a constant moves the center but leaves the spread untouched.
缩放与平移
- 线性变换 $aX + b$ 按 $a$ 缩放、按 $b$ 平移。
- 均值：$\mu_{aX+b} = a\,\mu_X + b$——缩放和平移都会传递过来。
- 标准差：$\sigma_{aX+b} = |a|\,\sigma_X$——缩放拉伸分散；平移不会。
- 加一个常数会移动中心，但让分散保持不变。
Means of sums and differences
- Means always add (or subtract), no conditions needed:
-
$$\mu_{X+Y} = \mu_X + \mu_Y \qquad \mu_{X-Y} = \mu_X - \mu_Y$$
- Expected values combine linearly, whether or not the variables are independent.
- This is the easy, always-true half of the rules.
和与差的均值
- 均值总是相加（或相减），无需任何条件：
-
$$\mu_{X+Y} = \mu_X + \mu_Y \qquad \mu_{X-Y} = \mu_X - \mu_Y$$
- 期望值线性地组合，无论变量是否独立。
- 这是法则中简单、永远成立的那一半。
SD of sums and differences
- Variances add for independent variables — even for a difference:
-
$$\sigma_{X \pm Y}^2 = \sigma_X^2 + \sigma_Y^2$$
- Then $\sigma_{X\pm Y} = \sqrt{\sigma_X^2 + \sigma_Y^2}$.
- Note the $+$: you add variances even when subtracting the variables.
和与差的标准差
- 对独立变量，方差相加——即使是差也一样：
-
$$\sigma_{X \pm Y}^2 = \sigma_X^2 + \sigma_Y^2$$
- 然后 $\sigma_{X\pm Y} = \sqrt{\sigma_X^2 + \sigma_Y^2}$。
- 注意那个 $+$：即使在相减变量时，你也是把方差相加。
Combining independent normals
- A sum or difference of independent normal random variables is also normal.
- Its mean and SD come from the rules above ($\mu$'s add/subtract; variances add).
- So you can find probabilities for the combined variable with the normal model.
- This makes many two-variable questions solvable in one clean step.
组合独立正态变量
- 独立正态随机变量的和或差仍是正态的。
- 它的均值和标准差来自上面的法则（$\mu$ 相加/相减；方差相加）。
- 于是你可以用正态模型求组合变量的概率。
- 这让许多两变量问题能在一个干净的步骤里解决。
Two traps. (1) When you subtract variables, you still add their variances ($\sigma_{X-Y}^2=\sigma_X^2+\sigma_Y^2$) — never subtract variances. (2) Never add standard deviations directly ($\sigma_X+\sigma_Y$ is wrong); add the variances, then square-root. Both rules require the variables to be independent.
两个陷阱。（1）当你相减变量时，仍然要相加它们的方差（$\sigma_{X-Y}^2=\sigma_X^2+\sigma_Y^2$）——绝不相减方差。（2）绝不要直接把标准差相加（$\sigma_X+\sigma_Y$ 是错的）；先加方差，再开平方根。这两条法则都要求变量独立。
$X$ (mean $10$, SD $3$) and $Y$ (mean $4$, SD $4$) are independent.
- Mean of $X-Y$: $10 - 4 = 6$.
- Variance: $\sigma_X^2 + \sigma_Y^2 = 9 + 16 = 25$ (add, even for a difference).
- SD of $X-Y$: $\sqrt{25} = 5$.
$X$（均值 $10$，标准差 $3$）与 $Y$（均值 $4$，标准差 $4$）独立。
- $X-Y$ 的均值：$10 - 4 = 6$。
- 方差：$\sigma_X^2 + \sigma_Y^2 = 9 + 16 = 25$（相加，即便是差）。
- $X-Y$ 的标准差：$\sqrt{25} = 5$。
A linear transformation: $\mu_{aX+b}=a\mu_X+b$ but $\sigma_{aX+b}=|a|\sigma_X$ (shifts don't affect spread). Means add/subtract always; for independent variables variances add ($\sigma_{X\pm Y}^2=\sigma_X^2+\sigma_Y^2$) — even for a difference. A combination of independent normals is normal.
线性变换：$\mu_{aX+b}=a\mu_X+b$ 但 $\sigma_{aX+b}=|a|\sigma_X$（平移不影响分散）。均值总是相加/相减；对独立变量方差相加（$\sigma_{X\pm Y}^2=\sigma_X^2+\sigma_Y^2$）——即便是差。独立正态变量的组合仍是正态的。
A combined normal variable · 组合后的正态变量
A sum/difference of independent normals is itself normal. · 独立正态变量的和/差本身也是正态的。
X has mean 10, Y has mean 4. Find the mean of X − Y. · X 均值为 10，Y 均值为 4。求 X − Y 的均值。
Means subtract: 10 − 4 = 6. · 均值相减：10 − 4 = 6。
X has SD 3, Y has SD 4, independent. Find the SD of X − Y. · X 标准差为 3，Y 标准差为 4，独立。求 X − Y 的标准差。
Variances add: 9 + 16 = 25, so SD = √25 = 5. · 方差相加：9 + 16 = 25，所以标准差 = √25 = 5。
When subtracting two independent random variables, you subtract their variances. · 在相减两个独立随机变量时，你要相减它们的方差。
Variances ADD even for a difference — never subtract them. · 即使是差，方差也相加——绝不相减。
For the transformation 2X + 5, the standard deviation becomes... · 对变换 2X + 5，标准差变成……
SD scales by |a| but the shift +5 doesn't affect spread. · 标准差按 |a| 缩放，而平移 +5 不影响分散。
To combine standard deviations of independent variables, you should add the SDs directly. · 要组合独立变量的标准差，应该直接把标准差相加。
Add the variances, then square-root — not the SDs. · 先加方差，再开平方根——而不是加标准差。