Probability Theory

Discrete Random Variables and Standard Models

Probability mass functions and the standard discrete families used to model counts and waiting times. · 7 steps

Continuous Random Variables and Densities

Describing continuous randomness by density and distribution functions and the main continuous families. · 7 steps

Transformations of a Single Random Variable

Finding the distribution of a function of a random variable via the distribution-function and change-of-variables methods. · 7 steps

Joint Distributions and Independence of Random Variables

Joint, marginal and conditional distributions for pairs and vectors of random variables. · 7 steps

Sums, Convolutions and Bivariate Transformations

Deriving the distribution of sums, quotients and other functions of two or more random variables. · 7 steps

Expectation, Covariance and Correlation

Linearity of expectation and second-moment structure for collections of random variables. · 7 steps

Conditional Expectation and Variance

Conditioning as a computational tool through the tower rule and the variance decomposition. · 7 steps

Generating Functions and Moments

Moment and probability generating functions as tools for moments, sums and distribution identification. · 7 steps

Inequalities, Laws of Large Numbers and the Central Limit Theorem

Asymptotic behaviour of averages and sums of independent random variables. · 7 steps

Exam Synthesis: Multi-Step Problems and Strategy

Selecting and chaining the right tools under exam conditions on mixed, multi-part problems. · 7 steps