This IB Maths AA HL Topic AHL 4.14 – Random Variables and Probability Density Functions resource develops students’ understanding of discrete and continuous random variables, expected value, and variance. It begins by reviewing discrete probability distributions and the conditions required for a valid model. Students learn how to calculate E(X), E(X²), Var(X) = E(X²) - [E(X)]², and the standard deviation of a discrete random variable.

The lesson then applies expected value to games of chance, with particular emphasis on defining the random variable as the player’s net gain and using E(X) to determine whether a game is fair. Students are introduced to continuous random variables and probability density functions, learning that probabilities are represented by areas under the curve and that the total area must equal 1. Practice includes finding unknown constants, calculating probabilities by integration, and determining the mode, median, mean, and variance of a continuous distribution. The resource also explores transformations of random variables using E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X). With clear formula summaries, worked examples, optional proofs, integration practice, fair-game applications, exam checklists, and fully worked solutions, this resource helps HL students build strong probability-distribution skills in line with IB Mathematics AA expectations.