A hungry cat is chasing a mouse through the ground-floor rooms of a house, as depicted below

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A hungry cat is chasing a mouse through the ground-floor rooms of a house, as
depicted below.
Kitchen Living Room
Dining Room Hallway

Until they meet (i.e. are in the same room at the sametime) the cat and mouse
move independently according to their respective discrete-time Markov chain dy-
namics.
Every 10 minutes, the mouse will Choose to either stay in its Current room with
probability 0.9, or moveto a uniformly chosen adjacent room.
At the same moments in time the cat will also decide to either stay in its current
room with probability 0.1, or move to a uniformly chosen adjacent room.
The cat eats the mouse as soon as they are both in the same room at the same
time.
Consider a new Markov chain (D,),s+o Corresponding to the room-distance between
the cat and the mouse after 107 minutes, which can take values O (if they are in
the same room), 1 (if they are in adjacent rooms) or 2 (otherwise).
(a) i) Show that the 1-step transition matrix for (D,)azo on state-spaceS =

{0O,1, 2} is

1 O O
P= 041 018 041 ,
0.045 082 0.135
iit) Decompose the state space §S into the set of transient states, T, and the

closed irreducible set(s) of recurrent states C,,/ = 1,...,”, for some

n> 1.

(b) Suppose that the cat and mouse last moved at 11:52am and at noon the cat
is in the kitchen and the mouse is in the hallway. What is the probability that
at 12:15pm the cat and mouse will be in adjacent rooms?

(c) Suppose that, just after they have both moved, the cat is in the living room
and the mouse is in the dining room. Calculate the expected further time (in
minutes) until the cat eats the mouse. oS
Hint: for suitably defined {M,}, you might consider a relation of the form

M = 1+ dojes PUM, 1 AB
‘ O (= B-
(d) Now suppose that the cat is not hungry and is chasing the mouse just for
fun. The 1-step transition matrix of (D,),+9 becomes
O 1 O
Poew = 041 O18 0.41
0.045 0.82 0.135
What is the largetime probability that the cat and mouse will be in adjacent
rooms?

Suppose that X,,...,X, is a random sample from the Geometric distribution with
parameter 8 € (0,1), Geo(@), with probability mass function
px(x; 6) = 611-0)", x € {1,2,...}.
Recall that E[X,] = 1/8.
(a) Suppose Y has a Negative Binomial distribution with parameters n € N and
p€ (0,1), NB(n, p), with probability mass function
. _ y-1 n yon
py(y:p) = 7-1)? (l-p)’"", ye{nantl.,...}.
Show that the probability generating function of Y Is
zp "
le) = (ae)
(b) Show using generating functions or otherwise that }-7_, X; ~ NB(n, 8).
(c) Show that the Geometric distribution is an exponential family and determine
the canonical statistic based on the above random sample of size n.
(d) Show that X,1 x, <3 is an unbiased estimator of (1 — 6)*, where for event A,
1, is the random variable defined as
1 ifweA
La(w) =
lw) f otherwise.
(e) For each t>n+2, andxe {1,...,t-—n— 1}, show that
n (~*~)
P (x10. =x| \ Xi= } = EH
(=1 n=)
Hint: notice that for each x € {1, 2,...},
{X 11x, <3 = x} = {X, = x,X2 = 3}.
(f) Hence or otherwise find the unique minimum variance unbiased estimator of
(1 — 6)?.
Hint: you may use without proof
t—n—1
1 t-x-—4 t—n)(t-n-1
—~ yx x _ (t=aj(t=n )
(1) — n—3 (t—1)(t — 2)