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Probability Theory

Random variables, distributions, expectation, limit theorems.

ບົດຮຽນ

Introductory Sample spaces and the axioms Kolmogorov's axioms, events as sets, conditional probability and Bayes. 1 - (5/6)^4 Core Random variables and expectation Distributions, expected value, variance, linearity of expectation. 10 * 1/2 Core The common distributions Bernoulli, binomial, Poisson, uniform, exponential, normal, and what each models. integrate e^(-x^2/2)/sqrt(2*pi) dx from -1.96 to 1.96 Advanced The law of large numbers and the central limit theorem Why averages converge and why they are normally distributed. limit of (1 + 1/n)^n as n -> oo

Statistics measures data; probability theory is the mathematics underneath it. Distributions as functions, expectation as an integral, and the laws that make averages settle down.

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Probability

An Introduction to ProbabilityConditional Probability and Independent EventsBernoulli TrialsDiscrete Random VariablesCentral TendencyProbability Spaces with Infinitely Many OutcomesDiscussion

Applying Probability to Combinatorics

A First Taste of Ramsey TheorySmall Ramsey NumbersEstimating Ramsey NumbersApplying Probability to Ramsey TheoryRamsey's TheoremThe Probabilistic MethodDiscussion

Foundations

Probability theoryProbability spaceProbability axiomsConditional probabilityBayes' theoremIndependence (probability theory)Random variableProbability distributionCumulative distribution functionProbability density function

Distributions

Binomial distributionPoisson distributionNormal distributionExponential distributionUniform distribution (continuous)Geometric distributionChi-squared distributionStudent's t-distribution

Limit theorems

Expected valueVarianceCovarianceLaw of large numbersCentral limit theoremChebyshev's inequalityMoment-generating functionConvergence of random variablesMarkov chainMartingale (probability theory)Stochastic processBrownian motion

ບົດຮຽນບາງບົດນີ້ ແມ່ນໄດ້ນຳເອົາຈາກບົດຂຽນເປີດ: Keller & Trotter, Applied Combinatorics, Wikipedia. ທຸກໆ​ໜ້າ​ບົດຮຽນ​ມີ​ຊື່​ແຫຼ່ງ​ຂອງ​ມັນ ແລະ ໃບອະນຸຍາດ​ຂອງ​ມັນ.

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What is the difference between probability and statistics?

Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.

What does the law of large numbers promise?

That the average of many independent samples converges to the expected value. It says nothing about any single trial.

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