Math 431 - Introduction to the Theory of Probability

• Prerequisites: Math 234.
• Frequency: Fall (I), Spring (II), Summer (SS)
• Student Body: Math majors and students in the Sciences and Engineering
• Credits: 3. (N-A)
• Recent Texts: A First Course in Probability, 6th Edition, by Sheldon Ross.
• Course Coordinator: Jim Kuelbs
• Background and Goals: no information available
• Alternatives: Math 331
• Subsequent Courses: Math 632, Math 831 course in probability. Topics to be covered include probability in discrete sample spaces, basic combinatorial analysis, conditional probability, stochastic independence, the Laplace limit theorem, Poisson approximation, random variables, laws of large numbers, the central limit theorem, applications.

Content coverage:

• Basic Concepts:
  • A mathematical model for a nondeterministic phenomenon
    • the sample space and events
    • probabilities of events
    • properties of probabilities
  • Finite sample spaces, equally likely outcomes, and methods of enumeration (combinatorics)
  • Conditional probability and independence
• Random Variables:
  • Definitions and important examples
    • the cumulative distribution function
    • expected value and variance
    • binomial random variables
    • Poisson random varaibles and approximation
    • geometric and negative binomial random variables
    • the uniform, normal, and exponential random vaariables
  • The distribution of a function of a random variable
  • Several random variables
    • definitions and examples
    • independent random varaibles
    • sums of independent random variables
    • conditional distributions
  • Further properties of expectations
    • properties of the expected value
    • properties of variances and covariances
    • conditional expectations and probabilities
• The Weak Law of Large Numbers and the Central Limit Theorem
  • Chebyshev's inequality
  • Proof of weak law of large numbers and applications
  • Application of central limit theorem to parameter estimation