Technical Notes

• The notes below are written in a concise form that is less readable than a textbook. They may be useful as condensed summaries but are not meant to be a replacement for a more complete explanation.
• Some of the notes were written as part of course material. As such they are not written in full generality and sometimes are not completely accurate. For example, measure-theoretic arguments are omitted and some assumptions such as differentiability are implicitly made.
• Please notify me if you find any mistakes or typos.

Probability

• Sample space, events, event space, probability as a function and its axioms. Discrete and continuous sample spaces
• Probability distributions on finite sample spaces, classical model for finite and continuous sample spaces
• Conditional probability, independence of events, Bayes theorem
• Random variables: basic definitions, discrete and continuous
• Important random variables: Bernoulli, geometric, binomial, Poisson, uniform, exponential, normal.
• Basic Combinatorics for Probability
• Functions of Random Variable
• Expectations and Variances
• Vector of Random Variables - basic definitions
• Functions of vector of random variables
• Conditional probability and vector random variables
• Expectation and vector random variables
• The multinomial and the multivariate Gaussian distributions
• The weak law of large numbers and central limit theorem
• The moment generating function
• The Poisson process

Statistics

• Sampling distributions
• The bias, variance and MSE of estimators
• Confidence intervals
• Relative efficiency, efficiency and the Fisher information
• Consistency of estimators
• Sufficient statistics
• The Rao-Blackwell theorem and the UMVUE
• The method of moments
• Maximum likelihood estimation
• Exponential family of distributions and logistic regression
• Hypothesis tests
• p-Values, power and the Neyman Pearson lemma
• Pearson's Chi-square
• Linear regression
• Inference in High Dimensions and Regularization
• Consistency of the maximum likelihood estimator
• Asymptotic Efficiency of the maximum likelihood estimator
• Basic sampling methods
• Markov chain Monte Carlo and Metropolis-Hastings
• Missing data and the EM algorithm
• Hidden Markov models
• Statistical Classifiers - the Generative Story
• Support vector machines
• Inference in Linear Regression
• M-estimators and Z-estimators
• Stein's Unbiased Risk Estimator
• Shrinkage and the James-Stein Estimator for Normal Means

Random Processes and LTI filtering

• Random processes - basic definitions
• Discrete-time discrete-valued random processes (the iid process and counting processes)
• Discrete-time discrete-valued random processes (the iid process and counting processes)
• Continuous-time discrete-valued random processes (Poisson process)
• Continuous-time continuous-valued random processes (Gaussian processes and the Wiener process)
• Stationary and wide sense stationary (WSS) processes
• Response of LTI systems to WSS processes
• Markov chains

Computer Science

• Turing Machines
• Complexity and Algorithms
• Operating Systems and Useful Linux Commands

Miscellaneous

• Convex Sets
• Convex Functions
• Connections and covariant derivatives
• A Quick Introduction to R
• Graphing One Dimensional Numeric Data