CE107: Mathematical Methods of System Analysis

Catalog copy
CE107.  Mathematical Methods of Systems Analysis. Introduction to
fundamental tools of stochastic analysis. Probability, conditional
probability, Bayes theorem, random variables, independence,
discrete-time stochastic processes, and Markov chains.  Instructor's
choice of additional topics, most likely drawn from confidence
measures, difference equations, transform methods, stability issues,
applications to reliability, queues, and hidden Markov models.


Explanation of prerequisites
CE16: students must be familiar with and proficient in set operations,
mathematical induction and basic notions of summation and recurrences.
Mathematics 24 or 27: students are expected to be proficient in first
 order linear differential equations.


Required Skills to pass the course
1. Know and apply probability axioms.
2. Understand notions of conditional probability and independence,
  Bayes' theorem, and be able to apply them to simple computer
  performance and reliability problems.
3.Compute moments of random variables.
4.Understand the notion of correlation, and be able to compute the
  first two moments of a sum of independent random variables, handle the
  maximum and minimum of independent random variables.
5.Be able to apply Markov's and Chebychev's inequality to simple
  bounding problems.
6.Use the Central Limit Theorem to estimate simple confidence intervals.
7. Understand the derivation of time-dependent equations of a
  one-dimensional birth and death process, derive and solve the
  steady-state equations.
8.Understand basic concepts of Markov chains, notion of
  ergodicity, and be able to obtain the stationary solution for a simple
  homogenous Markov chain.

Core topics (must be taught)

1.Probabilistic phenomena and relationship to experiments, notions of
  event, random variable, sample space, probability measures,
  probability axioms.
2.Conditional Probability, law of total probability, independence of
  events, Bayes' theorem, application to reliability
3. Distribution function, pmf, pdf (discrete/continuous random
  variables), moments.
4. Jointly distributed random variables, covariance, independence,
 sums of independent random variables, convolution, and conditional moments.
5. Markov's and Chebychev's inequalities and applications, law of
  large numbers.
6. Selected probability distributions & applications: binomial,
  geometric, Poisson, negative exponential, Gaussian random variable,
  Central Limit Theorem.
7. Elements of Stochastic Processes: basic notions, examples, Poisson
  process, birth and death process, equilibrium, steady state
8. Markov chains, state classification, ergodicity, simple applications.


Optional topics

1. Transform methods: moment generating function, generating function.
2. One-sided inequality, Bonferroni's inequality, Chernoff bound.
3. Erlang distribution, hypoexponential, hyperexponential, Gamma
   distribution.
4. Pareto distribution and application to network traffic characterization.


Text
R.D. Yates and D.J. Goodman, "Probability and stochastic processes: a
   friendly introduction for electrical and computer engineers", John
   Wiley & Sons, 1999.

Prepared by Alexandre Brandwajn, 04/02