HW5_Solutions - IE:3610 Stochastic Modeling(Fall 2015...

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IE:3610 Stochastic Modeling (Fall 2015) Homework 5 (due Thu, Oct 1) Solutions HW5 (due Thu, Oct 1): 29.2-1(a); 29.2-2(a); 29.3-1(a,b) 29.3-2(a,b); 29.3-3(a,b), 29.4-2, 29.4-5. 29.2-1(a) Solution: (a) Since the probability of rain tomorrow is only dependent on the weather today, Markovian property holds for the evolution of the weather. 29.2-2(a) Solution: (a) Let states to be: 1 = increased today and yesterday, 2 = increased today and decreased yesterday, 3 = decreased today and increased yesterday, 4 = decreased today and yesterday. P (1) = α 1 0 1 - α 1 0 α 2 0 1 - α 2 0 0 α 3 0 1 - α 3 0 α 4 0 1 - α 4 29.3-1(a,b) Solution: (a) as Chapman-Kolmogorov Equation: P (2) = P (1) * P (1) P (2) = 0 . 3 0 . 7 0 . 14 0 . 86 P (5) = 0 . 175 0 . 825 0 . 165 0 . 835 P (10) = 0 . 167 0 . 833 0 . 167 0 . 833 P (20) = 0 . 167 0 . 833 0 . 167 0 . 833 (b) P ( rain n days from now | rain today ) = P ( n ) 11 P ( rain n days from now | clear today ) = P ( n ) 21 1
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P (rain n days from now) = P ( rain n days from now | rain today ) * P ( rain today )+ + P ( rain n days from now | clear today ) * P ( clear today ) P (rain n days from now) = P ( n ) 11 * P ( rain today )+ P ( n ) 21 * P ( clear today ) If probability of rain today is 0 . 5: P (rain 2 days from now) = 0 . 3 * 0 . 5 + 0 . 14 * (1 - 0 . 5) = 0 .
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  • Spring '08
  • KROKHMAL
  • Ode, weather, Markov process, Rain, Ninja Assassin, 0.17 10 days, 0.22 5 days

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