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Events, the probability function and the Kolgomorov axioms
Conditional probability and Bayes' theorem
Entropy
Variables
Expected value, conditional expectation and Jensen's inequality
Variance and covariance
Higher moments
Markov's inequality and Chebyshev's inequality
Characteristic functions
Single observation discrete distributions
Simple continous distributions
Independent and identically distributed variables
The weak law of large numbers
Levy's continuity theorem
The central limit theorem and the gaussian/normal distribution
Statistics
Order statistics
Repeated observations discrete distributions
Extreme value distributions
Mixture distributions
Stochastic processes and their moments
White noise, and weak- and wide-sense stationarity
Random walks
Martingale processes
Markov processes
Multivariate time series
Bayesian networks
Survival functions
Orders of integration
Auto-Regressive processes, Moving-Average processes and Wold's theorem
Vector Autoregression (VAR)
ARMAX
Partial Adjustment Model (PAM)
Error Correction Model
Wiener processes and Brownian motion
Stochastic differential equations
Rejection sampling
Markov chain Monte Carlo sampling
Sampling from processes
Creating pseudo-random numbers
Stochastic methods for integration
Stochastic optimisation
Calculus of stochastic processes
Lossy compression
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