Roman C. answered • 11/18/15

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Masters of Education Graduate with Mathematics Expertise

(1) Q

_{ij}= P(X_{n+1}= j | X_{n}= i) = 1 2

1 ⌈ 0.9 0.1 ⌉

2 ⌊ 0.3 0.7 ⌋

(2) P(X

_{n+2}= 1 | X_{n}= 1)= P(X

_{n+2}= 1 | X_{n+1}= 1)P(X_{n+1}= 1 | X_{n}= 1) + P(X_{n+2 }= 1 | X_{n+1}= 2)P(X_{n+1}= 2 | X_{n}= 1)= 0.9

^{2}+ (0.3)(0.1) = 0.81 + 0.03 = 0.84(3) The long term probability for three consecutive weeks to all have high volume requires us to get the stationary distribution π

_{i}= P(X_{n}= i)πQ = π gives equations

0.9π

_{1}+ 0.3π_{2}= π_{1}, and π_{1}+ π_{2}= 1Solving this system gives π

_{1}= 0.75 and π_{2}= 0.25Now we are ready to answer the question.

P(X

_{n+3}= X_{n+2}= X_{n+1}= 1)= P(X

_{n+3}= 1 | X_{n+2}= 1)P(X_{n+2}= 1 | X_{n+1}= 1)P(X_{n+1}= 1 | X_{n}= 1)P(X_{n}= 1)+ P(X

_{n+3}= 1 | X_{n+2}= 1)P(X_{n+2}= 1 | X_{n+1}= 1)P(X_{n+1}= 1 | X_{n}= 2)P(X_{n}= 2)= 0.9

^{3}(0.75) + 0.9^{2}(0.3)(0.25)= 0.6075

Katie C.

11/18/15