Combining Random Variables

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:
• $$\mu_{X+Y} = \mu_X + \mu_Y \qquad \mu_{X-Y} = \mu_X - \mu_Y$$
• Expected values combine linearly, whether or not the variables are independent.
• This is the easy, always-true half of the rules.

For independent random variables means add and variances add

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.

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.

Write from a mean of the sum equal to 10

• Independent random variables have means 9 and 1.
• The lesson pair of means 6 and 5 is another problem.