Key Idea: A random variable has a mean (expected value) and a spread (variance). You find them, see how they change when you scale or add variables, and estimate them from a sample.
| Quantity | Rule |
|---|---|
| Mean | E(X) = Σ x·P(X=x) |
| Variance | Var(X) = E(X²) − [E(X)]² |
| SD | √Var(X) |
| Y = aX + b | E(Y) = aE(X) + b; Var(Y) = a²Var(X); SD(Y) = |a|·SD(X) |
Combining and estimating
- Independent X, Y: E(X±Y) = E(X)±E(Y), but Var(X±Y) = Var(X)+Var(Y). Variances always add.
- Sum of n independent copies: mean nμ, variance nσ². This is not Var(nX) = n²σ².
- x̄ estimates μ, and sₙ₋₁² (the GDC's Sx²) estimates σ². Both are unbiased.
Adding a constant moves the mean but never the spread.
E(X) = 1.70, Var(X) = 4.41 and G = X − 2. Find E(G) and SD(G). E(G) = −0.30 and SD(G) = 2.10.
Is that game fair to the player? No. On average the player loses 30p a spin.
Jar: mean 380 g, SD 4 g. Lid: mean 12 g, SD 1 g. SD of the total? √(16 + 1) = √17 ≈ 4.12 g.
Var(X) = 5. Find Var(3X + 1). 9 × 5 = 45.