Probability

Build the algebra, set notation, functions, summations, products, and logical statements used in probability calculations and proofs.

Count ordered and unordered outcomes with the addition and multiplication rules, permutations, combinations, and inclusion–exclusion.

Use probability measures, complements, unions, intersections, and countable additivity to derive probabilities from the Kolmogorov axioms.

Trace probability from games of chance and actuarial tables through Bayes, frequentist ideas, measure theory, stochastic processes, and computer simulation.

Restrict a sample space when information arrives, apply the multiplication rule, and solve inverse-probability problems with Bayes’ theorem.

Distinguish independent events from mutually exclusive events, test independence, and work with pairwise and mutual independence.

Represent numerical outcomes with random variables, probability mass functions, cumulative distribution functions, and indicator variables.

Calculate expected values, variance, standard deviation, moments, and covariance, including with indicator-variable shortcuts.

Model counts and categories with Bernoulli, binomial, geometric, negative binomial, hypergeometric, multinomial, and Poisson distributions.

Use limits, derivatives, integrals, improper integrals, and multivariable integration to analyze continuous probability models.

Work with densities, cumulative distribution functions, quantiles, mixed distributions, and probabilities over continuous ranges.

Model durations, measurements, extremes, and positive quantities with uniform, exponential, gamma, beta, normal, lognormal, Weibull, and related distributions.

Find the distribution of a transformed random variable using cumulative distributions, inverse transformations, and Jacobians.

Describe several random variables with joint, marginal, and conditional distributions, then calculate probabilities over multidimensional regions.

Use vectors, matrices, quadratic forms, eigenvalues, and positive semidefinite matrices to work with random vectors and covariance structures.

Analyze multivariate normal distributions, linear combinations, conditional Gaussian laws, and geometric probability regions.

Measure association with covariance, correlation, rank dependence, and tail dependence while recognizing that dependence need not be linear.

Separate marginal behavior from dependence using copulas, and construct multivariate models with Gaussian, Archimedean, and empirical copulas.

Compute conditional densities and conditional expectations, then apply total probability, total expectation, and total variance.

Treat probability as a representation of uncertain belief, update prior distributions into posteriors, and use exchangeability to model repeated observations.

Choose actions under uncertainty with loss functions, utility, expected value, value of information, and risk-sensitive decision criteria.

Use probability-generating functions, moment-generating functions, characteristic functions, and Laplace transforms to study sums and distributions.

Create reproducible probability experiments with pseudorandom generators, seeds, vectorized computation, numerical libraries, notebooks, and diagnostic plots.

Generate samples by inverse transforms, rejection sampling, importance sampling, and composition methods, then check whether a simulator matches its target law.

Estimate probabilities and expectations by Monte Carlo simulation, quantify simulation error, and improve efficiency with variance-reduction techniques.

Bound probabilities without knowing a full distribution using Markov, Chebyshev, Jensen, Hölder, Minkowski, and union inequalities.

Construct sigma-algebras, measurable functions, probability measures, and product spaces to give infinite and continuous models a rigorous foundation.

Define expectation through Lebesgue integration and work with almost-sure equality, absolute continuity, and the Radon–Nikodym derivative.

Treat conditional expectation as a random variable determined by available information, using sigma-algebras and filtration-based reasoning.

Distinguish almost-sure, probability, distribution, and mean convergence, and apply tools such as Fatou’s lemma and dominated convergence.

Show why sample averages stabilize, state weak and strong laws of large numbers, and recognize the assumptions each version requires.

Standardize sums, apply the central limit theorem, assess normal approximations, and use Berry–Esseen bounds to judge approximation error.

Control deviations of sums and functions of random variables with Chernoff, Hoeffding, Bernstein, Azuma, and bounded-difference inequalities.

Compare probability laws by constructing joint versions, use coupling to prove convergence and mixing, and measure distributional distance with Wasserstein metrics.

Model step-by-step random movement, recurrence, hitting times, absorption, and scaling behavior with one- and multidimensional random walks.

Analyze discrete-state systems through transition matrices, communicating classes, stationary distributions, reversibility, and mixing times.

Model random arrivals with Poisson processes, thinning, superposition, waiting-time laws, and nonhomogeneous intensities.

Analyze systems that restart after random intervals using renewal equations, long-run renewal rates, and reward processes.

Model congestion and service systems with birth–death chains, Little’s law, utilization, waiting times, and common queueing models.

Work with fair-game processes, filtrations, stopping times, optional stopping, and martingale convergence.

Construct Brownian motion and analyze its increments, paths, hitting times, reflection principle, and relationship to diffusion.

Define Itô integrals, apply Itô’s formula, and distinguish stochastic integration from ordinary calculus.

Model continuously evolving random systems with stochastic differential equations, then study existence, simulation, and long-run behavior.

Specify distributions over functions with means and kernels, condition Gaussian processes on observations, and use them for interpolation and uncertainty.

Analyze stationary processes through autocovariance, spectral density, ergodicity, linear filters, and probabilistic time-series models.

Model reproduction, epidemics, and population survival with Galton–Watson processes, extinction probabilities, and continuous-time branching.

Represent random locations and spatially varying quantities with point processes, marked processes, random fields, and spatial dependence.

Model maxima, minima, exceedances, and return levels with extreme-value limits and generalized Pareto tails.

Estimate very small probabilities using importance sampling, splitting, exponential tilting, and diagnostics designed for rare events.

Calculate failure probabilities, lifetimes, hazards, system reliability, and dependence among components in engineered and operational systems.

Quantify information with entropy, conditional entropy, mutual information, relative entropy, and data-processing inequalities.

Prove that combinatorial objects exist by random construction, first- and second-moment arguments, alteration, and the local lemma.

Use randomization to design and analyze sampling, hashing, streaming, approximation, and randomized search algorithms.

Study concentration in many dimensions, random projections, random matrices, spectral behavior, and phase changes that appear as dimension grows.

Represent conditional independence with directed and undirected graphs, factor joint distributions, and perform exact or approximate inference.

Sample difficult target distributions with Metropolis–Hastings, Gibbs sampling, Hamiltonian Monte Carlo, and convergence diagnostics.

Approximate changing distributions with particle filters, sequential importance sampling, resampling, and particle-based smoothing.

Express generative models as programs and use automatic differentiation and inference engines to fit, test, and compare them.

Infer model parameters when likelihoods are unavailable using approximate Bayesian computation, neural density estimation, posterior calibration, and simulation diagnostics.

Construct finite-sample predictive sets with conformal methods, assess coverage, and recognize the limits of distribution-free guarantees.

Propagate uncertain inputs through mathematical or computational models, separate sources of variation, and measure sensitivity with local and global methods.

Carry a real uncertainty problem from question and assumptions through model choice, analytic calculation or simulation, validation, stress testing, and communication of results.

Map probability skills to research, data science, finance, insurance, engineering, computing, and public policy; build proof and simulation portfolios and identify suitable advanced study paths.