Question

My cat spends his day sleeping for exactly one hour in different places in the living room, sometimes on the couch, sometimes on the table, and sometimes on the floor. When an hour-long nap is up, precisely when the clock strikes the hour, he either stays where he is or moves to a different location. If he’s on the floor, he’ll stay there 40% of the time, jump on the couch 30% of the time, and jump on the table 30% of the time. If he’s on the couch, he’ll stay there 80% of the time, jump down to the floor 20% of the time, but will never jump over to the table. If he’s on the table, he’ll stay where he is 50% of the time, jump onto the couch 30% of the time, and jump down to the floor 20% of the time.

a) Draw the transition matrix for the three states floor, couch, and table: F, C, T.

b) Draw a state diagram for that transition matrix.

c) It’s 8am and the kitty has just fallen asleep on the couch. Given that he is currently on the couch, what is the probability that he will be asleep on the couch just before the clock strikes 10am?

d) Each morning at 8am, I carry my cat downstairs and set him on the floor with probability π1 = 0.6, the couch with π2 = 0.3 and the table with π3 = 0.1. What is the probability that he will be asleep on the couch just before the clock strikes 10am?

Answer 1

Answers of above questions are given as:-

a) Transition table is given as

	F	C	T
F	0.40	0.30	0.30
C	0.20	0.80	0.00
T	0.20	0.30	0.50

That is the probability of moving a cat from floor to couch is 0.30 and to table is 0.30 and the probability of staying on the floor is 0.40 similarly for others.

b) state diagram is given as