In Problems 1-6, identify the absorbing states in the indicated transition matrix.
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- 1. The Galápagos penguin is the only penguin species found north of the equator, and it is considered endangered. The population of the Galápagos penguin changes on a yearly basis as a discrete dynamic [40] system. Suppose that initially there are 40 chicks and 40 breeding adults, that is xo = Suppose 40 also that the yearly transition matrix is A = = 0 [o. 0.125 2 0.75 (a) Which entry in the transition matrix gives the annual birthrate of chicks per adult? Which entry tells us the proportion of chicks that will survive to become adults? (b) Calculate the state vector x1. (c) Find the eigenvalues and corresponding eigenvectors for matrix A. (d) Express xo as a linear combination of eigenvectors. (e) Find the long-term distribution, after n years. State the long term ratio of chicks to adults. marks] (f) From your result to part (e), is the Galápagos penguin likely to survive?arrow_forwardMilan.sttarrow_forwardIn the Heidman’s Department Store problem of Unit 2 of this module, suppose that the following transition matrix is appropriate:If Heidman’s has $4000 in the 0–30-day category and $5000 in the 31–90-day category what is your estimate of the amount of bad debts the company will experience?arrow_forward
- 2. Draw a graph from the given adjacency matrix. 1 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1arrow_forwardQuestion 3 a. An analog communication system with random variable X has probability density function f(x) defined by kx", f(x) ={k(2-x²), -l1 Determine i. the value of k. ii. F(x) iii. o, iv. Sketch the graph of f(x) and F(x). b. Consider a communication channel where each substation transmits and receive data. The probability between the substations is shown in figure 3.arrow_forwardConsider a communication channel where each substation transmits and receive data. The probability between the substations is shown in figure 3. 0.4 0.5 0.3 0.2 2 2 0.3 Figure 3 i. Draw a transition diagram. ii. Write down a transition matrix, P. iii. Name the type of matrix in Q3(b) P(X, =2 X, = 1|X, =3) iv. P(X, =2 X, = 3X, =2) V. enarrow_forward
- 2.Initially we have 2 red marbles in Box 1 and 2 white marble in Box 2. At every time step, we take a marble from each box and swap. Let Xt be the number of red marbles in Box 1 at time t. There are 3 states: 0, 1, 2. Let P be the transition matrix for {X;}, what is (P)oo (probability from state 0 to state 0)?arrow_forwardLet us have the experience of a mouse falling into a trap inside a 6-room building, as shown in the figure below. If the room contains k doors, the probability that the mouse will choose one of these doors is 1/k if the mouse reaches room F that contains the food or room S that contains the trap. It will stay there and the experiment ends, the transition matrix is M= [0.5 0 0.5 0 0.3 0 0 ¹0.5 0 0 0.3 0.3 0 0.3 0.3 0 0.5 0 000 O a O b О с 2 000 0.5 0 0.5 0 0 0 0 0.3 0 0.5 0 0.5 M = - 0 0.5 0.5 b 3 ..+. 4 5 0 0 S M= 0 0 0 1 0 0.5 0 0 0 0 0.5 0 0 0.3 0.3 0 0 0.3 0 0.5 0 0 0.5 0 0 1 00000 10 0 000 0.5 0 0 0.5 0 0.3 0 0.3 0.3 0 0.3 0.3 0 0 0 0.5 0 0 000 0 0 0 0 C 0 0 0.3 0.5 1 (a)arrow_forwardIn some country, 90% of the daughters of working women also work and 20% of the daughters of nonworking women work. Assume that these percents remains 0.9 0.2 unchanged from one generation to the next. The corresponding transition matrix is A = 0.1 0.8 Consider a typical group of women, of whom 45% currently work. Use A and A² to determine the proportion of working women in the next two generations. The proportion of working women in the first generation is %. (Type an integer or a decimal.) %. The proportion of working women in the second generation is (Type an integer or a decimal.)arrow_forward
- Consider a communication channel where each substation transmits and receive data. The probability between the substations is shown in figure 3. 0.4 0.5 0.3 0.2 2 2 3 0.3 Figure 3 i. Draw a transition diagram. ii. Write down a transition matrix, P. iii. Name the type of matrix in Q3(b) P(X; =2 X,=1\X, =3) iv. P(X; =2 _X,=3{X, =2) V.arrow_forward2. Find the state transition matrix of the system whose state-space A = L- B = C = [0 1] -4.arrow_forward5. consider the example below, where the states are Condition State and the transition matrix is 0 1 23 2 Πο Good as new Operable-minimum deterioration Operable-major deterioration Inoperable and replaced by a good-as-new machine we found that the steady-state probabilities are 2 13' T1= = P = 78314 00 1 0 7 13' HARTNO LELBLINO 1 16 1 1 1 1 16 8 8 0 2 2 = π2 2 13' I3 = 2 13 (a) Find the expected recurrence time for state 0 (i.e., the expected length of time a machine can be used before it must be replaced) by solving a linear system for Moo, M10, 20, and μ30. (b) Find the expected recurrence time for state O directly by the formula Moo 1 πο =arrow_forward
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