MATH 1110 - Section 001
Spring 2012
Quiz 07 Name:
March 20, 2012
Directions: Write legibly, the use of documents is forbidden, giving or receiving aid on this assignement is forbidden. You will get zero if any of these happen.
0.1 0.6 0.8 0.1 0.1 1. Let A = 0.7 0.3 0.1, B = 0.3 0.3 0.2 0.2 0.1 0.6 0.6 Determine whether the following statements
0.1 0.3 1 0.3, C = 0.7 0 0.6 are true or false.
(a) A is a regular stochastic matrix. (b) B is a stable stochastic matrix. (c) C is a regular stochastic matrix. 4 points
Answer Statement 1 is false whereas statements 2 and 3 are true. 2. Each day is either rainy or sunny. If it rains one day, there is a 90% chance that it will be sunny the following day. If it is sunny one day, there is a 60% chance of rain the next day. In the long, run, what is the daily likelihood of rain ? (Let ’RAIN’ be state 1, and ’SUNNY’ be state 2). Detail your answer. 6 points
0.6 Answer The stochastic matrix is: 0.4 x The goal is to find the stable distribution X = . Solve the following system: y x + y = 1 x+y =1 x+y =1 0.1 0.6 x x 0.1x + 0.6y = x −0.9x + 0.6y = 0 → − → − = 0.9 0.4 y y 0.9x + 0.4y = y 0.9x − 0.6y = 0 Since the second third in this system it remains to solve: and equations areequivalent, 1 1 x 1 1 1 x 1 = [2] + 0.9[1] = −0.9 0.6 y 0 −−−−−−−→ 0 1.5 y 0.9 1 1 x 1 1 0 x 0.4 2 = [1] − [2] = 3 [2] 0 1 y 0.6 0 1 y 0.6 −−→ −−−−−→ x 0.4 The stable distribition is = , thus the daily likelihood of rain is 40%. y 0.6 0.1 0.9
MATH 1110 - Section 001
Spring 2012
Quiz 07 Name:
March 20, 2012
Directions: Write legibly, the use of documents is forbidden, giving or receiving aid on this assignement is forbidden. You will get zero if any of these happen.
0.1 0.6 0.8 0.1 0.1 1. Let A = 0.7 0.3 0.1, B = 0.3 0.3 0.2 0.2 0.1 0.6 0.6 Determine whether the following statements
0.1 0.3 1 0.3, C = 0.7 0 0.6 are true or false.
(a) A is a regular stochastic matrix. (b) B is a stable stochastic matrix. (c) C is a regular stochastic matrix. 4 points
Answer Statement 1 is false whereas statements 2 and 3 are true. 2. Each day is either rainy or sunny. If it rains one day, there is a 90% chance that it will be sunny the following day. If it is sunny one day, there is a 60% chance of rain the next day. In the long, run, what is the daily likelihood of rain ? (Let ’RAIN’ be state 1, and ’SUNNY’ be state 2). Detail your answer. 6 points
0.6 Answer The stochastic matrix is: 0.4 x The goal is to find the stable distribution X = . Solve the following system: y x + y = 1 x+y =1 x+y =1 0.1 0.6 x x 0.1x + 0.6y = x −0.9x + 0.6y = 0 → − → − = 0.9 0.4 y y 0.9x + 0.4y = y 0.9x − 0.6y = 0 Since the second third in this system it remains to solve: and equations areequivalent, 1 1 x 1 1 1 x 1 = [2] + 0.9[1] = −0.9 0.6 y 0 −−−−−−−→ 0 1.5 y 0.9 1 1 x 1 1 0 x 0.4 2 = [1] − [2] = 3 [2] 0 1 y 0.6 0 1 y 0.6 −−→ −−−−−→ x 0.4 The stable distribition is = , thus the daily likelihood of rain is 40%. y 0.6 0.1 0.9
