You Are NOT Allowed to Use Tables of Mathematical Formulas in the Exam

TEST 1 Sep 05

Time: 80 min. Lecturer : Armin Halilovic Allowed help: Calculator You are NOT allowed to use tables of mathematical formulas in the exam. Using of any communication device is strictly prohibited when taking this examination.

INSTRUCTIONS: Write your name and ID-number on each page.

Grading: For each correct solution 1 bonus point will be awarded. A maximum of 4 points can be earned. Every correct solved problem on tests gives you 1 bonus point that will be added on to the result on your final written examination. You can get maximum 4 points per test. Earned bonus points are valid one school year (until 31 Aug 2007).

0.3 0.3 x    Question 1. Let P 0.3 0.4 y  denote the transition probability matrix for a discrete 0.4 z 0.2 time Markov chain (Markov system) with three states E1 , E2 and E3 .  For the initial probability distribution vector p (0) =(0.2, 0.3, w)   i) Evaluate p (1) , p (2) ii) Find the probability that the system is in the state E2 with the time point t=2 iii) Draw the transition probability diagram

Question 2. Let r(t) denote the failure rate and F(t) survival probability (reliability). Show that t F(t)   r( x)dx . e 0

t b) A system has the failure rate r(t)  . 50 Find the life distribution function F(t) Question3 A system has   2 failures per time unit. Interval times between failures have exponential distribution. Reparation has a rate of   8 reparations per time unit. We model the system as a birth-and-death process with two states E0 and E1 We denote

p0 (t) = probability that the system works correctly at time point t , and

p1 (t) = probability that the system does not work correctly at time point t (and is under reparation)  We assume that the probability for the system to work correctly at time point t=0 is p0 (0)  0.2

(consequently p1 (0)  0.8). a) Find the rate transition matrix Q and determine E E 1 2 the system of differential equations for the p0 (t)

and p1 (t) .

b) Solve the system and find p0 (t) and p1 (t)  c) Find the probability that the system works correctly at time point t= 0.12 time units

Question4) A system can be modeled as a three state Markov chain. The system has three states E1, E2 and E3 with transition rates defined below in the rate transition diagram. a) Find the rate transition Q- matrix. b) Determine the stationary probability vector.

E 1 2 3

E E 3 1 6

Good luck! Answer till KS1:

0.3 0.3 0.4   1) a) P  0.3 0.4 0.3 0.4 0.4 0.2   p(1)  (0.35, 0.38, 0.27) , p(2)  (0.327, 0.365, 0.308) b) 0.365

2) a) f (t) F(t) t t F(x) r(t)   r(t)    r(x)dx   dx  F (t) 1 F(t) 0 0 1 F(x) t t  F(x)  r(x)dx   dx  0 0 1 F(x) t x  t  r(x)dx   ln |1 F(x) |  0 x  0 t  r(x)dx  ln(1 F(t))  0 t 1 F(t)   r(x)dx  e 0 t F(t)   r( x)dx e 0 b) t t x  r(x)dx  dx t2 /100 F(t)     50  e e 0 e 0 2 F(t)  1 F(t)  1 et /100

 2 2  3) a) Q     8  8 b) p0(t)  0.8  0.6e 10t p1(t)  0.2  0.6e 10t c) p0(0.12)  0.619

1 1 0     4) a) Q   0  3 3  b) p  (2 / 3, 2 / 9, 1/ 9)  6 0  6