Statistical Analysis and Simulation (ODU ECE 651)

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    Fundamental aspects of probability theory are covered, starting from basic concepts such as set theory, probability space, conditional probabilities, independence,  the law of large numbers, to random variables, joint and marginal densities, general moments, transformations, the central limit theorem, and finally, to more advanced topics such as queuing theory, linear prediction, LMS and Kalman filters . Special attention is given to random processes and power spectral estimation.

    Lecture at Old Dominion University  (Spring 2015)



    Text book:
    Papoulis, Pillai, Probability, Random Variables, and Stochastic Processes, Mc Graw Hill, 4th ed.
    1. The meaning of probability – Chapter 1
      • Definitions of probability: classical, relative frequency, axiomatic
    2. Fundamental concepts in probability – Chapter 2
      • Set theory
      • The probability space
      • Conditional probability and independence
    3. Repeated trials – Chapter 3 and Section 4
      • Combined experiments / Bernoulli trials
      • Asymptotic theorems
      • Law of large numbers
      • Rare events: Poisson theorem/random points
    4. The concept of a random variable – Chapter 4
      • Distribution and density functions
      • Conditional distributions and density functions
      • Total probability and Bayes’ theorem
    5. Functions of one random variable – Chapter 5
      • Functions of a random variable
      • Mean, variance and general moments
      • Characteristic functions
    6. Two random variables – Chapter 6
      • Joint and marginal statistics
      • Functions of two random variables
      • Joint moments and characteristic functions
      • Conditional distributions and expected values
    7. Sequences of random variables – Chapter 7
      • Transformations
      • Mean and covariance
      • Conditional densities and characteristic functions
      • The central limit theorem
    8. Random Processes – Chapter 9 and Section 12.1
      • Statistics of random processes
      • The independent increments property
      • Stationary processes: strict sense and wide sense
      • Systems with stochastic inputs
      • The power density spectrum
      • Ergodicity

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