# hw9 - 1 A Cost for each state State1=(0,1 so cost=2(0-1)^2...

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1. A) Cost for each state: State1=(0,1) so cost=2(0-1)^2+0=2 State2=(1,1), cost=2(1-1)^2+1=1 State3=(0,0), cost=2(0-0)^2+0=0 State4=(1,0), cost=2(1-0)^2+1=3 Transition probabilities: f(Cost_i, Cost_j, T)=1/2exp(-Cost/T) for Cost_ij>0 since 1/N(Si)=1/2=prob of moving to neighbor. f(1,1)=0, f(1,2)=1/2 as given, f(1,3)= 1/2 since Cost_ij<0, f(1,4)=0, f(2,1)= 1/2*exp(-1/T), f(2,2)=1 – 1/2*exp(-1/T) – 1/2 *exp(-2/T) f(2, 3)=0, f(2,4)= ½*exp(-2/T), f(3,1)= ½*exp(-2/T), f(3,2)=0 f(3,3)= 1 – 1/2*exp(-2/T) – 1/2 *exp(-3/T) f(3,4)= 1/2 *exp(-3/T), f(4,1)=0 f(4,2)=1/2=f(4,3) f(4,4)=0 There is no self loop in states 1 and 4 since they cannot be neighbors of themselves as the domain of each state is {0, 1}. B) P= [0, 1/2, 1/2, 0; 1/2*exp(-1/T), 1-1/2*exp(-1/T)-1/2*exp(-2/T), 0, 1/2*exp(-2/T); 1/2*exp(-2/T), 0, 1-1/2*exp(-2/T)-1/2*exp(-3/T), 1/2*exp(-3/T); 0,1/2,1/2,0] For T=1, P = 0 0.5000 0.5000 0 0.1839 0.7484 0 0.0677 0.0677 0 0.9074 0.0249 0 0.5000 0.5000 0 C) P^5 = 0.0981 0.3251 0.5407 0.0361 0.1196 0.4952 0.3412 0.0440 0.0732 0.1255 0.7744 0.0269

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0.0981 0.3251 0.5407 0.0361 P^10 = 0.0916 0.2725 0.6022 0.0337 0.1002 0.3413 0.5216 0.0369 0.0815 0.1919 0.6966 0.0300 0.0916 0.2725 0.6022 0.0337 P^1000 = 0.0871 0.2369 0.6439 0.0321 0.0871 0.2369 0.6439 0.0321 0.0871 0.2369 0.6439 0.0321 0.0871 0.2369 0.6439 0.0321 This means the probability distribution starts at a random state is [.0871, 0.2369, 0.6439, 0.0321] D) For T=10, P = 0 0.5000 0.5000 0 0.4524 0.1382 0 0.4094 0.4094 0 0.2202 0.3704 0 0.5000 0.5000 0 P^5 = 0.1281 0.3685 0.3875 0.1159 0.3334 0.1650 0.1999 0.3017 0.3173 0.1809 0.2148 0.2871 0.1281 0.3685 0.3875 0.1159 P^10 = 0.2771 0.2208 0.2514 0.2507 0.1998 0.2974 0.3220 0.1808 0.2059 0.2914 0.3165 0.1863 0.2771 0.2208 0.2514 0.2507 P^1000 = 0.2363 0.2612 0.2887 0.2138
0.2363 0.2612 0.2887 0.2138 0.2363 0.2612 0.2887 0.2138 0.2363 0.2612 0.2887 0.2138 This means the probability distribution starts at a random state is [0.2363, 0.2612, 0.2887, 0.2138] E) For T=0.2, P = 0 0.5000 0.5000 0 0.0034 0.9966 0 0.0000 0.0000 0 1.0000

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hw9 - 1 A Cost for each state State1=(0,1 so cost=2(0-1)^2...

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