# MAST30001 Tute2 Sol 2022.pdf - MAST30001 Stochastic...

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MAST30001 Stochastic ModellingTutorial Sheet 21. Show that the Markov property does not in general imply that for any eventB, andstatesi, j,P(Xn+1=j|XnB, Xn-1=i) =P(Xn+1=j|XnB).(Define a Markov chain, eventB, and statesi, j, where the equality doesn’t hold.)
2. Let (Yn)n1be i.i.d. random variables withP(Yi= 1) =P(Yi=-1) = 1/2 and letXn= (Yn+1+Yn)/2.(a) Find the “transition probabilities”P(Xn+m=k|Xn=j) form= 1,2, . . .andj, k= 0,±1.Hint: Separate the casesm= 1andm6= 1.(b) Show that (Xn)n1isnota Markov chain.Ans.
3. Let (Xn) be a Markov chain with state space{1,2,3}and transition matrix01/32/31/43/402/503/5(a) ComputeP(X3= 1, X2= 2, X1= 2|X0= 1).(b) IfX0is uniformly distributed on{1,2,3}, computeP(X3= 1, X2= 2, X1= 2).(c) Now assumingP(X0= 1) =P(X0= 3) = 2/5, computeP(X1= 2, X4=2, X6= 2).(d) AssumingP(X0= 1) = 1, use simulation to estimate the proportion of timethe chain spends in each state. Compare these proportions to the probabilitythat that the chain is in each state at time 100.Ans.

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Term
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Tags
yn, Markov chain, Random walk