A first course in probability and Markov chains /

• Κύριος συγγραφέας: Modica, Giuseppe
• Άλλοι συγγραφείς: Poggiolini, Laura
• Μορφή: Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Chichester : Wiley, 2013.
• Θέματα: Markov processes.; MATHEMATICS > Probability & Statistics > General.; Electronic books.

"Provides an introduction to basic structures of probability with a view towards applications in information technology A First Course in Probability and Markov Chains presents an introduction to the basic elements in probability and focuses on two main areas. The first part explores notions an...

Πίνακας περιεχομένων:

• Chapter 1 Combinatorics; 1.1 Binomial coefficients; 1.1.1 Pascal triangle; 1.1.2 Some properties of binomial coefficients; 1.1.3 Generalized binomial coefficients and binomial series; 1.1.4 Inversion formulas; 1.1.5 Exercises; 1.2 Sets, permutations and functions; 1.2.1 Sets; 1.2.2 Permutations; 1.2.3 Multisets; 1.2.4 Lists and functions; 1.2.5 Injective functions; 1.2.6 Monotone increasing functions; 1.2.7 Monotone nondecreasing functions; 1.2.8 Surjective functions; 1.2.9 Exercises; 1.3 Drawings; 1.3.1 Ordered drawings.
• 1.3.2 Simple drawings1.3.3 Multiplicative property of drawings; 1.3.4 Exercises; 1.4 Grouping; 1.4.1 Collocations of pairwise different objects; 1.4.2 Collocations of identical objects; 1.4.3 Multiplicative property; 1.4.4 Collocations in statistical physics; 1.4.5 Exercises; Chapter 2 Probability measures; 2.1 Elementary probability; 2.1.1 Exercises; 2.2 Basic facts; 2.2.1 Events; 2.2.2 Probability measures; 2.2.3 Continuity of measures; 2.2.4 Integral with respect to a measure; 2.2.5 Probabilities on finite and denumerable sets; 2.2.6 Probabilities on denumerable sets.
• 2.2.7 Probabilities on uncountable sets2.2.8 Exercises; 2.3 Conditional probability; 2.3.1 Definition; 2.3.2 Bayes formula; 2.3.3 Exercises; 2.4 Inclusion-exclusion principle; 2.4.1 Exercises; Chapter 3 Random variables; 3.1 Random variables; 3.1.1 Definitions; 3.1.2 Expected value; 3.1.3 Functions of random variables; 3.1.4 Cavalieri formula; 3.1.5 Variance; 3.1.6 Markov and Chebyshev inequalities; 3.1.7 Variational characterization of the median and of the expected value; 3.1.8 Exercises; 3.2 A few discrete distributions; 3.2.1 Bernoulli distribution; 3.2.2 Binomial distribution.
• 3.2.3 Hypergeometric distribution3.2.4 Negative binomial distribution; 3.2.5 Poisson distribution; 3.2.6 Geometric distribution; 3.2.7 Exercises; 3.3 Some absolutely continuous distributions; 3.3.1 Uniform distribution; 3.3.2 Normal distribution; 3.3.3 Exponential distribution; 3.3.4 Gamma distributions; 3.3.5 Failure rate; 3.3.6 Exercises; Chapter 4 Vector valued random variables; 4.1 Joint distribution; 4.1.1 Joint and marginal distributions; 4.1.2 Exercises; 4.2 Covariance; 4.2.1 Random variables with finite expected value and variance; 4.2.2 Correlation coefficient; 4.2.3 Exercises.
• 4.3 Independent random variables4.3.1 Independent events; 4.3.2 Independent random variables; 4.3.3 Independence of many random variables; 4.3.4 Sum of independent random variables; 4.3.5 Exercises; 4.4 Sequences of independent random variables; 4.4.1 Weak law of large numbers; 4.4.2 Borel-Cantelli lemma; 4.4.3 Convergences of random variables; 4.4.4 Strong law of large numbers; 4.4.5 A few applications of the law of large numbers; 4.4.6 Central limit theorem; 4.4.7 Exercises; Chapter 5 Discrete time Markov chains; 5.1 Stochastic matrices; 5.1.1 Definitions; 5.1.2 Oriented graphs.