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Lecture notes for the probability with engineering applications course (ece 313) offered at the university of illinois at urbana-champaign in fall 1997. The notes cover topics such as approaches to probability, conditional probability, independence, random variables, limit theorems, and decision making under uncertainty. The document also includes a syllabus, a table of contents, and references to additional resources.
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Hours Approaches to Probability 5 Subjective approach, classical approach, relative frequency approach, modeling issues; the axiomatic approach. Consequences of the axioms and examples, use of Venn diagrams and Karnaugh maps, principle of inclusion and exclusion. Conditional Probability 6 Definition of conditional probability, theorem of total probability, Bayes’ formula and its use. Bayes’ rule for deciding among competing hypotheses, maximum-likelihood (ML) rule, Type I and Type II errors Independence and Independent Trials 6 Stochastic independence of two events, Independence of multiple events, Reliability of systems and networks, Independent experiments and trials, Random Variables 12 Definition, Cumulative Distribution Function of a random variable: Discrete and continuous random variables: Mean and variance: mean, mode and median as measures of location, Markov’s inequality; variance, Chebyshev’s inequality and variance as a measure of spread. Examples of discrete and continuous distributions: Function of a random variable: expectation of a function of a random variable. Conditional Distributions: Reliability and Hazard rates: Hypothesis testing: Maximum-likelihood estimation of parameters of distributions: Many Random Variables 11 Joint distributions, covariance and correlation coefficient, jointly Gaussian random variables. Sums of random variables, other functions of many random variables, conditional distributions. Linear regression Limit Theorems 2 Markov’s Inequality, Chebyshev’s Inequality, Weak law of large numbers and central limit theorem Exams Total 44
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Preface................................................................................................................................................................... iii Syllabus................................................................................................................................................................. iv
