Informal Introduction to Stochastic Processes with Maple (Universitext)
Jan Vrbik, Paul Vrbik
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.
to n! (n being the size of the matrix).This makes the algorithm practical for small matrices only (in our case, no more than 4 ×4) and impossible (even when using supercomputers) for matrices beyond even a moderate size (say 30 ×30). INVERTING MATRICES (OF ANY SIZE) The general procedure (easy to code) requires the following steps: 1.Append the unit matrix to the matrix to be inverted (creating a new matrix with twice as many columns as the old one), for example, 2.Use any number of the
means (5.7) reduces to Differentiating three times yields which reduces to implying (since ). Replacing H ′′ (1) by , F ′′ (1) by , and G ′′ (1) by (where σ1 2 and σ2 2 are the individual variances of the number of trials to generate the first and second patterns, respectively), we get implying where P 1 (P 2) is the probability that the first (second) pattern wins the game. Example 5.9. When playing 2 consecutive sixes against 10 consecutive nonsixes, the previous formula yields
how much they buy), which explains why it is called a cluster. Using the first interpretation of Y j , we are interested in the total amount of money spent by those customers who arrived during the time interval (0, t), or The moment-generating function (MGF) of Y (t) is thus where M Y (u) is the MGF of each single purchase Y j . The expected value of Y (t) is simply λtμ Y (just differentiate the preceding expression with respect to u and evaluate at u = 0). Proposition 6.4. Proof. The
type of each of the four categories, namely: 1.Finite Markov chains, branching processes, and the renewal process (Chaps. 1–4); 2.Poisson process, birth and death processes, and the continuous-time Markov chain (Chaps. 5–8); 3.Brownian motion (Chaps. 9); 4.Autoregressive models (Chaps. 10). Solving such processes (for any finite selection of times t 1, t 2, …, t N ) requires computing the distribution of each individual X(t), as well as the bivariate distribution of any X(t 1), X(t 2)
and H. M. Taylor. A Second Course in Stochastic Processes. Academic, New York, 1981. 8. J. G. Kemeny and J. L. Snell. Finite Markov Chains. Springer, New York, 1976. 9. J. Medhi. Stochastic Processes. Wiley, New York, 1994. 10. J. Medhi. Stochastic Models in Queueing Theory. Academic, Amsterdam, 2003. 11. S. Ross. Stochastic Processes. Wiley, New York, 1996. 12. A. Stuart. Kendall’s Advanced Theory of Statistics. Wiley, Chichester, 1994. Jan Vrbik and Paul VrbikUniversitextInformal