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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.
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Probability
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[P1] |
Sample space, events,
event space, probability as a function and its axioms. Discrete
and continuous sample spaces |
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[P2] |
Probability
distributions on finite sample spaces, classical model for
finite and continuous sample spaces |
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[P3] |
Conditional probability, independence of events, Bayes theorem |
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[P4] |
Random variables: basic
definitions, discrete and continuous |
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[P5] |
Important random variables: Bernoulli, geometric, binomial,
Poisson, uniform, exponential, normal. |
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[P6] |
Functions of Random Variable |
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[P7] |
Expectations and Variances |
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[P8] |
Vector of Random Variables - basic definitions |
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[P9] |
Functions of vector of
random variables |
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[P10] |
Conditional probability and vector random variables |
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[P11] |
Expectation and vector
random variables |
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[P12] |
The multinomial and
the multivariate Gaussian distributions |
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[P13] |
The weak law of large numbers and central limit theorem |
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[P14] |
The moment
generating function |
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Statistics
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[S1] |
Sampling
distributions |
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[S2] |
The bias, variance
and MSE of estimators |
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[S3] |
Confidence intervals |
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[S4] |
Relative efficiency,
efficiency and the Fisher information |
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[S5] |
Consistency of
estimators |
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[S6] |
Sufficient
statistics |
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[S7] |
The Rao-Blackwell
theorem and the UMVUE |
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[S8] |
The method of moments |
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[S9] |
Maximum likelihood estimation |
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[S10] |
Hypothesis tests |
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[S11] |
p-Values, power and the Neyman Pearson lemma |
| [S13] |
Likelihood ratio test |
| [S14] |
Pearson's Chi-square |
| [S15] |
Linear regression |
| [S16] |
Consistency of the maximum likelihood estimator |
| [S17] |
Asymptotic Efficiency of the maximum likelihood estimator |
| [S18] |
Basic sampling methods |
| [S19] |
Markov chain Monte Carlo and
Metropolis-Hastings |
| [S20] |
Missing data and the
EM algorithm |
| [S21] |
Hidden Markov models |
| [S22] |
Support vector machines |
| [S23] |
M-estimators and Z-estimators |
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Random
Processes and LTI filtering
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[R1] |
Random processes - basic
definitions |
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[R2] |
Discrete-time discrete-valued random processes (the iid process
and counting processes) |
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[R3] |
Continuous-time discrete-valued random processes (Poisson
process) |
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[R4] |
Continuous-time continuous-valued random processes (Gaussian
processes and the Wiener process) |
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[R5] |
Stationary and wide sense
stationary (WSS) processes |
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[R6] |
Response of LTI systems to WSS processes |
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Computer Science
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[C1] |
Turing Machines |
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[C2] |
Complexity and Algorithms |
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[C3] |
Operating Systems and Useful Linux Commands |
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Miscellaneous
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[M1] |
EM for means
of mixture of Gaussians |
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[M2] |
Convex Sets |
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[M3] |
Convex
Functions |
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[M4] |
Connections and
covariant derivatives |
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