Mathematics: Analysis and Approaches HL

Statistics and probability

Higher-level probability reasoning, Bayes theorem, and continuous random variables.

Use Bayes theorem and conditional probability Apply conditional probability and Bayes theorem in structured probability contexts. Use continuous random variables Use density functions, cumulative probability, expectation, and intervals for continuous variables. Use expected value and variance of discrete variables Calculate and interpret expectation, variance, and standard deviation for discrete variables. Use probability generating functions Use generating functions to extract probabilities, expectation, and distribution information. Use moment generating functions Extract moments and interpret distributions using moment generating functions. Use covariance and correlation of random variables Calculate and interpret covariance, correlation, and linear association for random variables. Use bivariate distributions Calculate and interpret probabilities from joint distributions. Use conditional expectation Calculate and interpret conditional expectation in discrete or continuous contexts. Use transformations of random variables Find distributions, means, or probabilities after transforming random variables. Use normal approximations Apply and interpret normal approximations with appropriate conditions and corrections. Use joint probability density functions Calculate and interpret probabilities from joint density functions. Find marginal and conditional distributions Derive marginal and conditional distributions from a joint model. Use linear combinations of random variables Calculate means and variances for linear combinations of random variables. Apply the central limit theorem Use the central limit theorem to approximate sampling distributions. Use covariance matrices Interpret covariance matrices and relationships between random variables. Analyse correlation under linear transformations Determine how linear transformations affect correlation, mean, and variance. Use moment relationships for distributions Use raw and central moments to interpret distribution behaviour. Approximate distributions with continuity corrections Apply continuity corrections when approximating discrete distributions. Use conditional variance Calculate and interpret conditional variance in probability models. Apply law of total expectation Use total expectation to combine conditional components. Use transformations of continuous variables Transform continuous random variables and interpret resulting densities or probabilities. Analyse sampling distribution approximations Use sampling distribution approximations and interpret assumptions.