Random Variables
A random variable converts outcomes into numbers so that probabilities can be handled algebraically. The exam tests the probability mass function, the probability density function, the distribution function, and simple probabilities computed from each.
Core rules
- A discrete random variable takes countable values with a pmf p(x) satisfying p(x) ≥ 0 and Σp(x) = 1.
- A continuous random variable has a pdf f(x) with f(x) ≥ 0 and ∫f(x)dx = 1 over its range. Density is not probability: f(x) may exceed 1, but P(X = x) = 0 for every x.
- For continuous X, P(a < X < b) = ∫f(x)dx from a to b.
- The distribution function F(x) = P(X ≤ x) is non-decreasing, right-continuous, tends to 0 at −∞ and 1 at +∞, and P(a < X ≤ b) = F(b) − F(a).
Worked example
A discrete X has P(X = 0) = k, P(X = 1) = 2k, P(X = 2) = 3k. Since probabilities sum to 1, 6k = 1, so k = 1/6. Then P(X ≥ 1) = 5/6. Also E(X) = 0(1/6) + 1(2/6) + 2(3/6) = 8/6 = 4/3, and E(X²) = 0 + 2/6 + 12/6 = 7/3, so V(X) = 7/3 − 16/9 = 5/9.
Traps that cost marks
Treating f(x) as a probability and reporting a value above 1 as an error; forgetting to check that probabilities or the density integrate to 1; mixing the discrete rule P(X ≤ b) − P(X ≤ a−1) with the continuous rule; computing variance as E(X²) − E(X) instead of E(X²) − [E(X)]².