Download Exercise set 8

MET270 Introduction to Statistics and Probability 2, autumn 2010.
Exercise set 8
Textbook (Ghahramani):
Section 12.3: Exercise 1, 3, 7
Section 12, review problems: Exercise 1, 3, 4
Exercise 1:
Suppose that the queue to be served at Idsø (a famous butcher in Stavanger) on a
Saturday is a M/M/4 queue with arrival rate λ = 1.5 per minute and expected service
time of 1/γ = 2 minutes. When you arrive four persons are being served and 10 are
waiting in line.
Suppose now that the chance of rain on a day does not only depend on the weather the
previous day, but also on the weather two days ago. Then we can not use the simple
Markov model above, but we can actually still model the situation as a Markov chain by
augmenting the state space to contain the weather sequence the two last days (see also
example 12.11 in Ghahramani). Let state 0: ss (sun-sun), state 1: rs (rain-sun), state 2:
sr (sun-rain), state 3: rr (rain-rain). Let {Xn : n = 0, 1, 2, . . .} be a stochastic process
with the four states above as state space.
Suppose the chance of rain is 0.3 if it was not raining the two previous days, 0.5 if it was
not raining the day before but was raining two days before, 0.6 if it was raining the day
before but not two days before and 0.8 if it was raining the two previous days.
b) What is the expected waiting time until you are starting to get served? What is the
expected waiting time until you are finished?
c) Explain why {Xn : n = 0, 1, 2, . . .} is
matrix
⎛
0.7
⎜
⎜ 0.5
P =⎜
⎜ 0
⎝
0
c) What is the probability that you are finished before the person in front of you in the
queue? What is the probability that the person behind you are finished before you?
It is given that:
a) Explain why the queue is stable. Would the queue still be stable if one of the four
persons serving customers had to leave work?
⎛
Suppose that the chance of rain tomorrow only depends on whether it is raining today
and not on past weather conditions. Let the states be labelled, 0: no rain, 1: rain.
Suppose that:
P (rain tomorrow|rain today) = 0.75
P (rain tomorrow|not rain today) = 0.35
⎞
0 0.3 0
⎟
0 0.5 0 ⎟
⎟
0.4 0 0.6 ⎟
⎠
0.2 0 0.8
0.49 0.12
0.20
0.12
0.10 0.16
⎜
⎜ 0.35
P2 = ⎜
⎜ 0.20
⎝
Exercise 2:
a Markov chain with transition probability
0.21
0.15
0.20
0.10
⎞
0.18
⎟
0.30 ⎟
⎟
0.48 ⎟
⎠
0.64
d) It is raining today, but was not raining yesterday, what is the probability that it
is not raining tomorrow? If it was not raining today and yesterday, what is the
probability it will be raining both the two next days? If it was raining yesterday,
but not today, what is the probability it will be raining tomorrow but not the day
after?
a) Find the transition probability matrix. Show that
P2 =
0.51 0.49
0.35 0.65
Some answers:
12.3:1 Not Markov; 12.3:3 a) 3/8, b) 11/16; 12.3:7 Recurrent: {0, 3} and {2, 4}. Transient: {1}.
b) If it is raining today, what is the probability that it is not raining tomorrow? If it
is not raining today, what is the probability that it will be raining in two days? If
it was raining yesterday, what is the probability that it will not be raining tomorrow?
12.review:1 0.692; 12.review:3 Rename 1 to 4, 3 to 1 and 4 to 3. 12.review:4 5/3.
1 a) no; b) 5.5 and 7.5; c) 3/8 and 3/8;
2 b) 0.25, 0.49 and 0.35; d) 0.4, 0.18, and 0.20.