Random Variables and Mathematical Expectation
Core rules
- A discrete random variable takes countable values with probability mass function p(x), where Σp(x) = 1.
- A continuous random variable has density f(x) ≥ 0 with total area 1; P(X = a) = 0 for any single point.
- Mean E(X) = Σx p(x) for discrete variables; variance Var(X) = E(X²) − [E(X)]².
- E(aX+b) = aE(X)+b; Var(aX+b) = a²Var(X).
- For continuous variables, replace sums by integrals.
- The cumulative distribution F(x) = P(X ≤ x) is non-decreasing from 0 to 1.
Worked example
X takes 0, 1, 2 with probabilities 0.2, 0.5, 0.3. E(X) = 0(0.2)+1(0.5)+2(0.3) = 1.1. E(X²) = 0²(0.2)+1²(0.5)+2²(0.3) = 0.5+1.2 = 1.7. Var(X) = 1.7−1.1² = 1.7−1.21 = 0.49. SD = 0.7.
Traps
- Writing E(X²) as [E(X)]²; they differ by the variance.
- Forgetting to check that probabilities sum to 1.
- Giving a negative variance from an arithmetic slip.
- Adding variances when variables are not independent.
- Treating f(x) as a probability rather than a density.