STATS320, Assignment questions, Semester 1 2023
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This assignment contributes7.5%of your final grade.
Maximum possible marks: 30 marks
1 A Markov chain (Xn, n= 0, 1 , 2 ,.. .) with state spaceS={ 1 , 2 , 3 , 4 , 5 }has the transition matrix
P=
0 .6 0. 3 0 0. 1 0
0 .5 0. 2 0 0 .2 0. 1
0 0. 7 0 0. 3 0
0 0 .4 0.4 0.1 0. 1
0 0 0 0 1
.
aDraw the transition diagram for this Markov chain.
b FindP(X 3 = 1|X 0 = 4, X 1 = 2, X 2 = 3).
c FindP(X 1 = 3, X 2 = 2|X 0 = 4).
d Find the two-step transition probabilityp(2) 35 =P(X 2 = 5|X 0 = 3).
e Findh 31 =P(Xn= 1 for somen|X 0 = 3), the probability of ever reaching state 1 starting
from state 3.
f Suppose the chain starts with initial distribution
P(X 0 = 1) =
1
2
, P(X 0 = 2) = 0, P(X 0 = 3) = 0, P(X 0 = 4) =
1
2
, P(X 0 = 5) = 0.
(i) FindP(X 1 = 2)
(ii) FindP(X 0 = 1|X 1 = 2).
(iii) AreX 0 andX 1 independent? Please justify your answer.
[20 marks]
2 Alex has three green balls (numbered 1 to 3) and three blue balls (numbered 4 to 6), plus a bin
with space for exactly 3 balls. At each step:
- He roll a fair 6-sided die and find the ball with the corresponding number.
- If that ball is currently in the bin, remove it and replace it with a ball of the opposite colour. Otherwise, do nothing.
LetXnto be the number of green balls in the bin afternsteps. Then (Xn, n= 0, 1 , 2 ,.. .) is a
Markov chain with state spaceS={ 0 , 1 , 2 , 3 }. Start with all the green balls in the bin and all the
blue balls outside, so thatX 0 = 3.
aPart of the transition matrix is given below. Find the remaining entries, explaining how you
obtained them.
P=
0 1/3 1/2 1/ 6
0 0 1 /2 1/ 2
b Find the equilibrium distribution.
c Is the limiting distribution the same as the equilibrium distribution? Justify your answer.
[10 marks]
1