Midterm 00-3 - People.stat.sfu.ca

STAT 280 Test Nov. 15, 2000
Instructions: The test is open book and notes. You have 90 minutes to complete the test.
Attempt all questions. The test marks total 90 - allocate your time so you can attempt all
the questions.
1. (30 marks) For each of the phenomena described below, suggest a probability model -
be sure to suggest specific parameter values. You do not need to justify your model
choice or choice of parameters unless you think your choice may be misunderstood. The
parameter values must be reasonable for the context described to obtain full marks.
a) The uncertainty in the game outcome when two equally-matched soccer teams
compete.
Bernoulli p=0.5 if ties are not allowed, or discrete (L,T,W) with prob .4 .2 .4 .
b) The variability in the average of the seven digits taken from the serial number of
randomly selected one-dollar bills.
Normal (mean=4.5, SD=1.1)
c) The distribution of life-length of ten taxis that are new today. (Assume that the taxis
are on the road constantly and the usual "death" of the taxi is a traffic accident which
renders the taxi unrepairable.)
Sample of n=10 from exponential distribution with mean of 3 years
d) A random walk with a slight tendency to increase over time.
Y= sum of iid Xs where X=+1 or - 1 with probability .55and .45 respectively.
e) The variability in the proportion of A grades (A-, A, and A+) in a class of 86
students, from one offering of the course to the next, assuming that the gradeassignment
process perfectly reflects the performance of each student.
The proportion should behave like a random sample from a 0-1 population. The
proportion of ones in the population would be about 0.2.
f) The actual rate at which students are served by the bookstore during the first 5
minutes of the semester.
There are at least 4 servers, and each customer would take about 1 minute on
average, so that would suggest a rate of 4 per minute. But this is only the average rate.
The actual rate would be described by dividing the Poisson (mean=20) by 5.
g) Imagine a bank closing at 5 pm, and at that time there are five persons lined up to be
served by a single teller. What model would you suggest for the length of time until
the teller has completed these services?
Gamma (shape parameter 5, rate parameter 1).
h) In ten tosses of a fair coin, the variability in the number of heads from its expected
value.
The Bin(n=10, p=.5) minus 5.
i) The number of cars parked in the three spaces beside the SFU Information office
during 10:00am - 11:00am today.
M/M/3 queue with the arrival rate of 15 per hour and a service rate of 15 per hour
for each space, and a capacity of 3.
j) The number of 6/49 lottery tickets one has to buy to win the jackpot. (Approx to
parameter value are OK).
Geometric p=1/14,000,000 approx

2. (15 marks) Fatal traffic accidents in the City of Vancouver occur at the rate of 2 per
month. Making whatever simplifying assumptions are reasonable and useful, compute
the following (Show your calculations, or otherwise explain how you arrived at your
numerical answers.).
a) The probability that exactly two accidents occur in the next 30 days.
Poisson mean 2 P(X=2)=.27
b) The probability that at least one accident occurs in the next 30 days.
1-.135=.865.
c) The average number of days until three more accidents occur.
Mean of Gamma(3,2) = 3/2 months or 45 days.
d) Given that exactly three accidents have occurred so far this month, what is the
probability that no more accidents will occur this month.
.135 (as in b, because of independent increments)
e) The probability that more than 24 accidents occur in the next 12 months.
.5 by normal approximation (bit less if continuity correction used)
3. (18 marks) Students leaving the SFU Bennett Library are classified as turning left,
right or proceeding across the mall towards the Maggie Benston Building. There is the
occasional student who is not thinking where they want to go, and instead simply follow
the direction of the person in front of them.
a) Construct a Markov Chain to describe this sequence of students
Let state space be L,S,R for Left, Straight, and Mall.
1 step trans matrix is .45 .20 .35
.35 .30 .35
.35 .20 .45
b) Compute the proportion of students turning right, assuming your Markov Chain model
to be correct.
.45 x1 + .35 x2 +.35 x3 = x1
.20 x1 + .30 x2 + .20 x3 = x2
x1 + x2 + x3 = 1
which produces x1=x3=.39 x2=.22 (rounded) So x3=.39 is the required proportion.
(For marking, if the equations are right, and the answer reasonable, accept as correct.)
c) Suppose you observe a student who chooses to go across the Mall. Suggest three
ways to compute the average number of steps in the Markov Chain until another student
crosses the Mall. (No calculations necessary, just write enough to indicate that you know
the methods.)
i) simulation
ii) Theorem 4.6-4 and use high power of 1-step transition matrix
iii) Theorem 4.6-4 and Theorem 4.6-3
ii) and iii) could be deemed to be the same method, so I guess give full
marks for two ways (bonus for three?).

4. (16 marks)
Customers arrive at a service desk at the rate of 20 per hour. There are three employees
serving customers at the service desk, and each one takes an average of four minutes to
serve a customer. Making whatever simplifying assumptions are reasonable and useful to
answer the following questions:
a) How often will all the servers be busy?
Compute PI0=.29 and PI3=.115 using formulas on pp 309-310
b) How would you calculate (exactly, not with simulation) the average amount of time a
customer will spend at the service desk (waiting and being served) before his service is
completed?
Use Little's formula on p 312. LambdaA is 20 and E(N) can be computed from
the PI0, PI1, PI2, and PI3 as in part a).
5. (11 marks)
An M/M/2 queuing system is a model for a service desk. A simulation of the model
produces the following results for the first five customers:
Customer arrivals services
1 2.90 2.71
2 1.24 1.67
3 0.49 1.85
4 1.34 2.27
5 0.50 7.59
The "arrivals" column records the interarrival time in minutes between customer n-1 and
customer n. (2.90 is the time from the start of the process until the first customer arrives.)
The "services" column gives the service time for each customer. Draw a graph of the
queue length for the first 10 minutes the service desk is open. (The graph can be
freehand, but must be accurate enough to show that you know how to use the data given
to construct the queue-length graph).
5. ANS.
An accurate graph is shown below. Of course the hand-drawn one would be less
precise, but should have the major features for full marks.

Further note to marker:
Concerning the queue graph:
The program produces this:
Tn Dn
1 2.90 5.61
2 4.14 5.81
3 4.63 7.46
4 5.97 8.24
5 6.47 15.05
3
2
1
0
0
the starting and finishing times for each arrival. Here is a detailed
verification of this.
5
Time
The queue jumps up to 1 at 2.90 and 2 at 4.14, and person #3 can't start
until #1 finishes at 5.61. So at 4.63, the queue jumps to 3. At 5.61, #1 completes and # 3
starts, dropping the queue to 2. ( So #3 finishes at 5.61 + 1.85 = 7.46.) At 5.81 #2
completes so the queue drops to 1
until # 4 arrives at 5.97, when it jumps back up to 2. When #5 arrives at 6.47, the queue
jumps up to 3 since #3 and #4 are still being served. At 7.46,
#3 completes and # 5 service starting then, so the queue drops to 2. At 8.24,
# 4 completes and the queue drops to 1 and stays there past the 10 minute mark.
I think my graph shows this fairly accurately. When I did a finer grid (1000
points instead of 100) it shows the same pattern but with more abrupt rises and falls.
10
15

Concerning the queue graph:
The program produces this:
Tn Dn
1 2.90 5.61
2 4.14 5.81
3 4.63 7.46
4 5.97 8.24
5 6.47 15.05
the starting and finishing times for each arrival. Here is a detailed
verification of this.
The queue jumps up to 1 at 2.90 and 2 at 4.14, and person #3 can't start
until #1 finishes at 5.61. So at 4.63, the queue jumps to 3. At 5.61, #1 completes and # 3
starts, dropping the queue to 2. ( So #3 finishes at 5.61 + 1.85 = 7.46.) At 5.81 #2
completes so the queue drops to 1
until # 4 arrives at 5.97, when it jumps back up to 2. When #5 arrives at 6.47, the queue
jumps up to 3 since #3 and #4 are still being served. At 7.46,
#3 completes and # 5 service starting then, so the queue drops to 2. At 8.24,
# 4 completes and the queue drops to 1 and stays there past the 10 minute mark.
I think my graph shows this fairly accurately. When I did a finer grid (1000
points instead of 100) it shows the same pattern but with more abrupt rises and falls.