D.5 Higher-Level Probability and Continuous VariablesHL

HL-only extension to D.3 (probability) and D.4 (distributions). Bayes' theorem flips a conditional: P(A | B) = P(B | A) · P(A) / P(B), letting you update a prior belief once you see new evidence. The IB caps Bayes problems at THREE mutually exclusive priors. The shortcut formula Var(X) = E(X²) − [E(X)]² avoids squaring deviations one-by-one. Linear transformations Y = aX + b shift the mean by +b and scale the variance by a² (the +b drops out — translations don't change spread). CONTINUOUS random variables replace probability mass with probability DENSITY: P(X = c) = 0 for any single point, and probability lives in the AREA under a pdf f(x) where ∫ f(x) dx = 1. Calculate mean μ = ∫ x·f(x) dx, median m via ∫_{−∞}^m f(x) dx = 0.5, mode by setting f'(x) = 0, and Var(X) = ∫ x²·f(x) dx − μ². Throughout: integration from E.2 is required for Lesson 3.