📚 Syllabus
Unit 1: Basic Background
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Basic Counting
Introduction to counting principles, sum rule, and historical context.
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Set Theory
Sets, operations, Venn diagrams, De Morgan's laws, and partitions.
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Sum and Product Rule
Fundamental counting principles with examples.
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Tuples
Ordered pairs, tuples, and Cartesian products.
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Combinations and Permutations
Permutations, combinations, and binomial coefficients.
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Practice Problems
Exercises and problems for Unit 1.
Unit 2: Probability
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Motivation for Probability
Introduction to probability, sample spaces, events, and axioms.
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Game Theory
Applications of probability in game theory.
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Random Walks
Introduction to random walks and their applications.
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Information Theory
Entropy, information, and coding theory basics.
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Conditional Probability
Conditional probability, Bayes' theorem, and independence.
Unit 3: Random Variables
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Random Variables
Introduction to random variables, discrete and continuous.
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Distributions
Common probability distributions: Bernoulli, Binomial, Poisson.
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Variance
Expectation, variance, and standard deviation.
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Moments
Moment generating functions and higher moments.
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Practice Problems
Exercises on discrete random variables.
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Cumulative Distribution Function
CDF properties and applications.
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Continuous Random Variables
PDF, expectation, and variance for continuous RVs.
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Solved Examples
Step-by-step solutions to key problems.
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Continuous Distributions
Uniform, Exponential, Normal distributions.
Unit 4: Multiple Random Variables
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Mixed Random Variables
Functions of random variables and transformations.
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Joint Distributions
Joint PMF, marginal distributions, and independence.
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Joint Continuous Random Variables
Joint PDF, covariance, and correlation.
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Multiple Random Variables
Multivariate distributions and properties.
Unit 5: Limit Theorems & Bounds
Unit 6: Statistical Inference
Unit 7: Random Processes
Content coming soon...