MAKAUT Discrete Mathematics Organizer | PCC-CS-401

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• Probability Theory & Distributions

  • Fundamental Concepts and Theorems: Defines axiomatic probability, Bayes' Theorem, and conditional probability formulations.
  • Common Probability Distributions: Covers Discrete (Binomial, Poisson) and Continuous (Uniform, Exponential, Normal) distributions alongside their expected values, means, and variances.
  • Limit Theorems: Explains the Central Limit Theorem and Chebyshev's (Tchebycheff's) Inequality for approximating probabilities of random variables.

• Mathematical Statistics & Hypothesis Testing

  • Estimation Theory: Examines point and interval estimation, maximum likelihood estimators (MLE), and properties like unbiasedness and consistency.
  • Sampling & Testing: Details sampling distributions, null/alternative hypotheses, Type I and Type II errors, $z$-tests, $t$-tests, and Chi-square ($\\chi^2$) goodness-of-fit tests.

• Advanced Graph Theory

  • Planar Graphs & Duals: Discusses planarity conditions, Euler's formula ($n - e + f = 2$), Kuratowski's graphs, and rules for constructing dual graphs.
  • Coloring & Eulerian Paths: Details graph coloring, chromatic numbers, chromatic polynomials for trees/circuits, and conditions for Eulerian circuits.

• Group Theory

  • Group Structures: Defines groups, abelian/commutative groups, subgroups, normal subgroups, quotient groups, and cyclic groups.
  • Group Theorems & Mappings: Explores Lagrange's theorem, symmetric/dihedral groups, permutation groups, and mappings including homomorphisms, isomorphisms, and kernels.

• Rings & Fields

  • Algebraic Systems: Defines rings, subrings, ideals, integral domains, zero divisors, and fields.
  • Key Properties: Establishes that every finite integral domain is a field, cancellation laws hold in rings without zero divisors, and field equations possess unique solutions.