TAILIEUCHUNG - Ebook Probability random variables and stochastic processes (4th edition): Part 2

(BQ) Approximately 1/3 of the text is new material - this material maintains the style and spirit of previous editions. In order to bridge the gap between concepts and applications, a number of additional examples have been added for further clarity, as well as several new topics. | 370 PROBABILITY AND RANDOM Variables 8-37 Show that in the measurement problem of Example 8-9. he bayesian estimate Ỏ of the parameter 0 equals à - ĩio. J. lgị1 . ơ aồ 9 yvo TX where Ơ X Ĩ -2 ơ0 Ơ II ƠỈ Ơ- II 8-38 Using the ML method find the y confidence interval of the variance V ơ of an A n. Ơ random variable with known mean. 8-39 Show that it fi and 8 are two unbiased minimum variance estimators of a parameter 9. then 81 9-. Hint Form the random variable 9 fl 9 2. Show that or o- l r 2 Ơ . where Ơ is the common variance of 91 and 9 and r is their correlation coefficient. 8-40 The number of successes of an event ri in II trials equals x- . Show that X l - lip. X. - np _ Ắ- - npi Pl lip 1 PiPi where 2 n - X and P A p 1 - p . 8-41 Let represent an unbiased estimator for the unknown parameter ip 9 based on the random variables X x . x X under joint density function f x 9 . Show that the Cramer-Rao lower bound for the parameter satisfies the inequality Var Tlx PART II STOCHASTIC PROCESSES 371 CHAPTER 9 GENERAL CONCEPTS 9-1 DEFINITIONS As we recall a random variable X is a rule for assigning to every outcome Ỉ of an experiment 5 a number x c . A stochastic process x t is a rule for assigning to every c a function x r Ị . Thus a stochastic process is a family of time functions depending on the parameter Ị or. equivalently a function of I and Ị. The domain of Í is the set of all experimental outcomes and the domain of t is a set R of real numbers. If R is the real axis then x r is a continuous-time process. If R is the set of integers then x r is a discrete-time process. A discrete-time process is. thus a sequence of random variables. Such a sequence will be denoted by x as in Sec. 7-4 or. to avoid double indices by x n . We shall say that x r is a discrete-stale process if its values are countable. Otherwise it is a continuous-state process. Most results in this investigation will be phrased in terms of continuous-time processes. Topics dealing with discrete-time

• Toán học
• Probability random variables and stochastic processes
• Probability random variables
• Stochastic processes
• Probability variables
• Stochastic processes
• Random variables
• Queueing system
• nite number
• shop assistants
• a forming of queues
• Queueing systems
• without queues
• Some general types
• stochastic processes Download free ebooks at bookboon
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• Probability Examples c 9 – Stochastic Processes 2 © 2009 Leif Mejlbro & Ventus Publishing ApS ISBN 978 87 7681 525 7 Download free ebooks at bookboon
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• Stochastic Processes 2 Contents Contents Introduction 1 1
• 1 1
• 2 1
• 3 1
• 4 1
• 5 1
• 6 1
• 7 2 3 4 5 Theoretical background The Poisson process Birth and death processes Queueing theory in general Queueing system of innitely many shop assistants Queueing system of a nite number of shop assistants
• and with forming of queues Queueing systems with a nite number of shop assistants and without queues Some general types of stochastic processes The Poisson process Birth and death processes Queueing theory Other types of stochastic processes Index 5 6 6 8 11 11 12 15 17 19 37 52 117 126 The next step for top performing gradua
• Discrete time stochastic processes
• Markov chains
• Continuous time stochastic processes
• Itô’s lemma
• Stochastic integral
• Preliminaries
• Lecture Introduction to Stochastic Processes
• Introduction to Stochastic Processes
• Disjoint time intervals
• The stochastic process
• Homogeneous Poisson proces
• Markov chain
• Discrete parameter stochastic process
• Time homogeneous Markov chain
• Lecture Probability Theory
• Probability Theory
• Lý thuyết xác suất
• Stationary Stochastic Processes
• Systems with Stochastic Inputs
• Homogeneous poisson processes
• Holds trivially
• Probability density function
• Order statistics property
• Conditional distribution
• Theorem superposition
• Probability generating function
• Poisson process
• stochastic calculus
• mathematics of finance
• Process Limits
• Invariant measur
• Unique stationary distribution
• Symmetric transition probability matrix
• SLLN for Markov chains
• The long run proportion
• Dissection principle
• Limiting distribution
• Aperiodic Markov chain
• Stationary distribution
• Homogeneous Poisson process
• Discrete Markov chains
• Positive real numbers
• Heuristic justification
• Rigorous justification
• Positive random variables
• Exponential distribution
• Poissonian increments
• Inhomogeneous Poisson process
• Discrete parameter Markov chain
• Countable state space
• Mathematical description
• Transition probabilities
• Simple random walk
• On off process
• Random walks on groups
• Gambler's ruin problem
• Chapman Kolmogorov equation
• Decomposition of state space
• Deterministically monotone Markov chain
• Communication classes
• The dissection principle
• Non empty closed subsets
• Recurrent Markov chain
• E valued random variables
• Power series
• Abel's limit theorem
• Proof of the theorem