MATH2750: Exercises 2
Examples 2.1 (Gambling Model). In the gambling model of Example 1.4, what is the probability that the game will go on indenitely, i.e. neither of the gamblers will ever be ruined? A. To answer the question, set Ri := P (0 < Xn < a +
8
MATH2750: Continuous time MPs
8.1
8.1.1
General processes with discrete states
Denition; Chapman Kolmogorov equations; intensities
Denition. Let state space of the process be S = cfw_1, 2, . . . or S = cfw_1, 2, . . . , N . Let
(i,j 0, i, j S, i = j). T
3
MATH2750
3.1
Stochastic process: r.vs which depend on TIME
Any stochastic process is, by denition, a collection of random variables depending on
time, i.e.
Xn , n = 0, 1, . . . , or Xt , t 0.
In most formulas we drop dependence of the r.v.s of an outcom
7
MATH2750: Continuous time MPs
7.1.1. Definition This is a continuous time analogue of the "discrete time jump process" from Handout 2.2, with S = 0, 1, 2, . . . By definition, X0 = 0. Then, the construction is as follows. There is a parameter > 0 called
4
4.1
MATH2750
Roulette (gambling): equation on a nite interval
As we already know from the solution of the Exercise 1.4, the gambling process Xn of
the fortune of the rst (say) player is a MP, with transition probabilities,
pi,i+1 = p,
pi,i1 = q = 1 p fo
1
1.1
MATH2750 - MARKOV PROCESSES, INTRO
Programme
Grading: 90% exam, 10% practicals (weeks 16, 19)1 . For the programme see pp 30-31 at http : /www.maths.leeds.ac.uk/school/students/Level%202%20modules%2020089.pdf For the recommended literature see http
5
5.1
MATH2750
Stationary (equilibrium) distributions
The notion of stationary distributions relates to a Markov chain understood as a family of processes with the same generator, rather than one process, each of which may be started from another initial
MATH2750: Introduction to Markov Processes
Exercises 5
E5.1. Consider a Markov chain on the state space S = cfw_1, 2, 3 with
transition matrix
1
1
1
3
3
3
0
0
P = 0
1
1
0
(a) Draw the transition graph of this chain.
(b) Show that state i = 1 is positive
MATH2750
MATH275001
This question paper consists of 4
printed pages, each of which is
identied by the reference MATH2750.
Only approved basic scientic
calculators may be used.
UNIVERSITY OF LEEDS
MockExamination for the Module MATH2750
Introduction to Mar
MATH2750
MATH275001
This question paper consists of 3
printed pages, each of which is
identied by the reference MATH2750.
Only approved basic scientic
calculators may be used.
UNIVERSITY OF LEEDS
MockExamination for the Module MATH2750
Introduction to Mar