AP Statistics / Unit 2: Probability, Random Variables, and Probability Distributions
2.9 Parameters of Random Variables
Notes
- Mean (expected value): $\displaystyle \mu_X=\sum x_iP(x_i)$.
- Standard deviation: $\displaystyle \sigma_X=\sqrt{\sum(x_i-\mu)^2P(x_i)}$.
- Linear changes: $\mu_{a+bX}=a+b\mu_X$, $\sigma_{a+bX}=|b|\sigma_X$.
- Sums of independent variables: means add; variances add.
Practice
$X$: values 0, 10 with probabilities 0.6, 0.4. $E(X)=$
$0(0.6)+10(0.4)$.
$\sigma_X=3$. $\sigma_{2X+5}=$
Adding 5 doesn't change spread; multiply by 2.
For independent $X$ and $Y$, which add?
$\sigma^2_{X+Y}=\sigma^2_X+\sigma^2_Y$.
Ready for more? Take the Unit 2 test.