Informal Introduction to Stochastic Processes with Maple (Universitext)
Jan Vrbik, Paul Vrbik
Language: English
Pages: 287
ISBN: 1461440564
Format: PDF / Kindle (mobi) / ePub
The book presents an introduction to Stochastic Processes including Markov Chains, Birth and Death processes, Brownian motion and Autoregressive models. The emphasis is on simplifying both the underlying mathematics and the conceptual understanding of random processes. In particular, non-trivial computations are delegated to a computer-algebra system, specifically Maple (although other systems can be easily substituted). Moreover, great care is taken to properly introduce the required mathematical tools (such as difference equations and generating functions) so that even students with only a basic mathematical background will find the book self-contained. Many detailed examples are given throughout the text to facilitate and reinforce learning.
Jan Vrbik has been a Professor of Mathematics and Statistics at Brock University in St Catharines, Ontario, Canada, since 1982.
Paul Vrbik is currently a PhD candidate in Computer Science at the University of Western Ontario in London, Ontario, Canada.
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the fixed probability vector of the following TPM: Also, find (in exact fractions) $$ {\lim \limits_{n\rightarrow \infty }} \mathbb{P}^{3N + 1}$$ Exercise 2.7. Find the fixed probability vector of Starting in State 1, what is the probability of being in State 4 after 1,001 transitions? Exercise 2.8. Calculate exactly using fractions: Exercise 2.9. Do a complete classification of Exercise 2.10. Do a complete classification of Exercise 2.11. Do a complete classification of
absorption during that particular transition, namely: The corresponding expected value, say , is given by analogously to . Since and are closely related, we can actually simplify the preceding formula using the following proposition. Proposition 3.1. Let be the column vector of row sums of , that is, the vector whose ith row/entry is given by . Then where and are two compatible matrices. Proof. We have This implies Proposition 3.2. τ can also be computed as the unique solution
compute The frequency of visits to State n is the corresponding reciprocal, namely, p n ⋅(λ n + μ n ). Example 7.3. Consider a linear growth with immigration (LGWI) process with individual birth and death rates both equal to 0. 25/h, an immigration rate of 0. 9/h, and an initial value of five natives. Find and plot the distribution of X(1. 35 h). Solution. 7.6 M ∕ M ∕ ∞ Queue M ∕ M ∕ ∞ denotes a queueing system with infinitely many servers, an exponential service time (for each server),
Throughout this book we use many commands that are contained in libraries. Since it would be cumbersome to call them in every worksheet, we assume the following packages are loaded at all times (the library names are case sensitive): 1.LinearAlgebra 2.Statistics 3.plots Lists and Sequences Sometimes we might want to consider a list or sequence of values. A list is an ordering of many values associated with a single name. The individual elements can be retrieved using “[ − ]” or by a
is, to start counting from i until a certain condition is met: This is useful when we want to stop on a condition but also require a counter to keep track of what step we are at. A few loop tips: 1.Unless you want to see the output for each step of the loop, be sure to close your loop with “end do:” not “end do;”. 2.In the worksheet, to get a new line without executing, do shift+return. 3.If you accidentally execute a loop that will never terminate (an infinite loop), then type ctrl+c or
