Combining Random Variables
Introduced| English |
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| linear transformation/ˈlɪnɪə trænsfɔːˈmeɪʃn/ |
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.
Means of sums and differences
- Means always add (or subtract), no conditions needed:
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$$\mu_{X+Y} = \mu_X + \mu_Y \qquad \mu_{X-Y} = \mu_X - \mu_Y$$
- Expected values combine linearly, whether or not the variables are independent.
- This is the easy, always-true half of the rules.
SD of sums and differences
- Variances add for independent variables — even for a difference:
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$$\sigma_{X \pm Y}^2 = \sigma_X^2 + \sigma_Y^2$$
- Then $\sigma_{X\pm Y} = \sqrt{\sigma_X^2 + \sigma_Y^2}$.
- Note the $+$: you add variances even when subtracting the variables.
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.
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.
$X$ (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 $X-Y$: $\sqrt{25} = 5$.
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.
A combined normal variable
A sum/difference of independent normals is itself normal.
X has mean 10, Y has mean 4. Find the mean of X − Y.
Means subtract: 10 − 4 = 6.
X has SD 3, Y has SD 4, independent. Find the SD of X − Y.
Variances add: 9 + 16 = 25, so SD = √25 = 5.
When subtracting two independent random variables, you subtract their variances.
Variances ADD even for a difference — never subtract them.
For the transformation 2X + 5, the standard deviation becomes...
SD scales by |a| but the shift +5 doesn't affect spread.
To combine standard deviations of independent variables, you should add the SDs directly.
Add the variances, then square-root — not the SDs.