Probability and Statistics
Probability and statistics supply the language for variability, and the exam tests whether you can pick the right model and avoid classic misreadings.
Distributions
Binomial: n independent trials with constant p; mean np, variance np(1−p). Poisson: counts per unit time or space; mean equals variance equals λt. Normal: use z = (x − μ)/σ. Exponential: time between Poisson events, memoryless, mean = standard deviation = 1/λ. Weibull: shape β < 1 gives decreasing hazard (infant mortality), β = 1 constant hazard, β > 1 increasing hazard (wear-out). Lognormal fits right-skewed quantities such as repair times.
Inference
Central Limit Theorem: the sample mean is approximately normal with standard error σ/√n - the standard error, not σ, is the denominator in confidence intervals and tests. Confidence interval for a mean: x̄ ± t·s/√n (z when σ is known and n is large). Hypothesis tests: α = P(Type I error) = rejecting a true null; β = P(Type II error) = failing to reject a false null; power = 1 − β. A p-value is the probability of data at least this extreme assuming H0 is true - it is not the probability that H0 is true.
Traps
Independence versus mutual exclusivity; "at least one" problems (use 1 − P(none)); dividing by √n when the question is about a single observation; one-sided versus two-sided intervals; a high R² does not prove causation and a regression slope is not a correlation coefficient.
Worked example
Twenty parts, each 2% defective, independent. P(at least one defective) = 1 − 0.98^20 = 1 − 0.6676 = 0.332. Subtracting 20 × 0.02 is the most common wrong answer.