Probability Mass Function (PMF)

Definition: $P(X = x)$ for each possible value $x$

Requirements:

• $0 ≤ P(X = x) ≤ 1$ for all $x$
• $\sum P(X = x) = 1$ (probabilities sum to 1)

Example: Number of heads in 3 fair coin flips

Check: $1/8 + 3/8 + 3/8 + 1/8 = 1$ ✓

Expected Value (Mean)

$E(X) = \mu_X = \sum x · P(X = x)$

Interpretation: Long-run average value

Example: $E(X) = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8) = 0 + 3/8 + 6/8 + 3/8 = 12/8 = 1.5$

Intuition: On average, 1.5 heads in 3 flips

Variance and Standard Deviation

$\text{Var}(X) = \sigma_X^2 = \sum (x - \mu)^2 · P(X = x) = E(X^2) - [E(X)]^2$

$\sigma_X = \sqrt{\text{Var}(X)}$

Example: For 3 coin flips ($\mu = 1.5$)

$\text{Var}(X) = (0-1.5)^2(1/8) + (1-1.5)^2(3/8) + (2-1.5)^2(3/8) + (3-1.5)^2(1/8)$ $= 2.25(1/8) + 0.25(3/8) + 0.25(3/8) + 2.25(1/8) = 0.75$ $\sigma_X = \sqrt{0.75} ≈ 0.866$

Linear Combinations of Random Variables

If $W = aX + b$: $E(W) = aE(X) + b$ $\text{Var}(W) = a^2 \text{Var}(X)$

If $Y = X_1 + X_2$ (independent): $E(Y) = E(X_1) + E(X_2)$ $\text{Var}(Y) = \text{Var}(X_1) + \text{Var}(X_2)$

If $Y = X_1 - X_2$ (independent): $E(Y) = E(X_1) - E(X_2)$ $\text{Var}(Y) = \text{Var}(X_1) + \text{Var}(X_2)$ (subtract in means, add in variances!)

Common Mistakes

• Forgetting variance is additive even for differences ($+ \text{Var}(X_2)$, not minus)
• Using $\text{Var}(X_1 + X_2) = \text{Var}(X_1) + \text{Var}(X_2)$ when variables are dependent
• Confusing PMF with CDF

Decision Rule

Modify random variable? → Apply linearity rules Independent sums? → Add means and variances

AP Exam Tip

"Find the probability distribution of X" means: create table with all possible values and their probabilities. Always verify probabilities sum to 1.