Exam - Probability and Stochastic models
Part A -
Q1. Twelve recruits were subjected to a selection test to ascertain their suitability for a certain course of training. At the end of training there were given a proficiency test.
The marks secure by recruits in the selection test (x) and the proficiency test (Y) are given below:
|
x:
|
65
|
63
|
67
|
64
|
68
|
62
|
70
|
66
|
68
|
67
|
69
|
71
|
|
y:
|
68
|
66
|
68
|
65
|
69
|
66
|
68
|
65
|
71
|
67
|
68
|
70
|
Calculate coefficient of rank correlation.
Q2. A study of the amount of rainfall and quantity of air pollution removed produced the following data:
|
Daily rainfall x (0.01 cm):
|
4.3
|
4.5
|
5.9
|
5.6
|
6.1
|
5.2
|
3.8
|
2.1
|
7.5
|
|
Particulate Removed y:
|
126
|
121
|
116
|
118
|
114
|
118
|
132
|
141
|
108
|
a) Make a scatter plot to the given data.
b) Find the equation of the regression line to predict the particulate removed from the amount of daily rainfall.
c) Estimate the amount of particulate removed when the daily rainfall is x-4.8 units.
Q3. Consider a certain community in well-defined area with three types-of grocery stores: for simplicity we shall call them I, II and III. Within this community (we assume that the population is fixed) there always exists a shift of customer from one grocery store to another. A study was made on January 1 and it was found that ¼ shopped at store 1, 1/3 at store II and 5/12 at store III. Each mouth store I retains 90% of its customers and loses 10% them to store II retains 90% of its customers and loses 50% of them to store I and 10% of the store III. Store III retains 40% of its customers and loses 50% of them to store I and 10% to store II.
What proportion of customers will each store retain by February 1, March 1 and April 1?
Q4. A research is analyzing brand switching between different airlines, operating on the Delhi-Mumbai route by frequent fliers. On the basis of the data collected by her, the researcher has developed the following transition probability matrix:
It is found that currently the airlines AA, BB and CC have 20%, 50% and 30% of the market respectively.
i) Obtain the market share for each airline in two moths time, and
ii) Obtain the market share for each airline in Four months time.
Q5. Continuous arrive to a hair cut salon according to a Poisson process with a mean arrival rate of 5/hr. Because of the reputation of the salon, customers were always willing to wait. Customer processing time was exponentially distributed with an average of 10 min. Answer the following questions:
a) Average no. of customers in the shop and average no. of customers weighting for hair cut.
b) Average no. of waiting when there is at least one person waiting.
c) How much time customers spend weighting in the queue?
d) What is the average system waiting time?
e) What is the probability that line delay is wit than 45 min.
Part B -
Q1. The following are the numbers of minutes it took 10 mechanics to assemble a piece of machinery in the morning x, and in the late afternoon, y:
|
x:
|
11.1
|
10.3
|
12.0
|
15.1
|
13.7
|
18.5
|
17.3
|
14.2
|
14.8
|
15.3
|
|
y:
|
10.5
|
14.2
|
13.8
|
21.5
|
13.2
|
21.1
|
16.4
|
19.3
|
17.4
|
19.0
|
Calculate r.
Q2. The following are measurement of the velocity and evaporation coefficient of burning fuel droplets in an impulse engine:
|
Air velocity (cm/s) x
|
20
|
60
|
100
|
140
|
180
|
220
|
260
|
300
|
340
|
380
|
|
Evaporation coefficient (mm2) y
|
0.18
|
0.37
|
0.35
|
0.78
|
0.56
|
0.75
|
1.18
|
1.36
|
1.17
|
1.65
|
a) Make a scatter plot.
b) Fit a straight line to these data by the method of least squares.
c) Use (b) to estimate the evaporation coefficient of a droplet when the air velocity is 190 cm/s.
Q3. Two manufactures A and B are competing with each other in a restricted market. Over the year, A's customers have exhibited high degree of loyalty as measured by the fact that customers using A's product 80 per cent of the time. Also former customers purchasing the product from B have switched back to A's 60% of the time.
a) Constructed the transition matrix.
b) Calculate the probability of a customer purchasing A's B's product at end of second period.
Q4. A police car is on patrol in a neighborhood known for its gang activities. During patrol, there is a 60% chance of responsibility in time to the location where help is needed, else regular patrol will continue. Upon receiving a call, there is a 10% chance for cancellation (in which case normal patrol is resumed) and a 30% chance that the car is already responding to a previous call. When the police car arrives at the scene, there is a 10%chance that the investigation will have filed (in which case the car returns back to patrol) and a 40% chance that apprehension occurs is made immediately. Else the offices will search the area. If apprehension occurs, there is a 60% chance of transporting the suspects to the police station, else they are released and the car returns to patrol.
i) Express the probabilistic activities of the police patrol in the form of transition matrix.
ii) If the police car is currently at a call scene, determine the probability that an apprehension will take place in two patrols.
Q5. The men's department of a large store employs one tailor for customer fittings. The number of customers requiring fittings appears to follow a Poisson distribution with mean arrival rate 24 per hour. Customers are fined on a first-come, first served basis, and they are always willing to wait for the tailor's service, because alternations are free. The time it takes to fit a customer appears to be exponentially distributed, with a mean of 2 min.
i) What is the average number of customer in the fitting room?
ii) How much time a customer is expected to spend in the fitting room?
iii) What percentage of the time is the tailor idle?
Part C -
Q1. The following data were obtained in a study of the relationship between the weight and chest size of infants at birth:
|
Weight (kg):
|
2.75
|
2.15
|
4.41
|
5.52
|
3.21
|
4.32
|
2.31
|
4.30
|
3.71
|
|
Chest size (cm):
|
29.5
|
26.3
|
32.2
|
36.5
|
27.2
|
27.7
|
28.3
|
30.3
|
28.7
|
Calculate the correlation coefficient.
Q2. Fit a straight line to the following data by the method of least squares and also predict moisture content when humidity is 56
|
Humidity x:
|
42
|
35
|
50
|
43
|
48
|
62
|
31
|
36
|
44
|
39
|
55
|
48
|
|
Moisture content y:
|
12
|
8
|
14
|
9
|
11
|
16
|
7
|
9
|
12
|
10
|
13
|
11
|
Q3. Suppose there are two market products of brand A and B respectively. Let each of these two brands have
|
|
To
|
|
A
|
B
|
|
From
|
A
|
0.9
|
0.1
|
|
B
|
0.5
|
0.5
|
If the initial market share breakdown is 50% for each brand, then determine their market shares in the next 6 time periods.
Q4. On January 1 (this year), Bakey A had 40% of its local market share while the other two bakeries B and C had 40% and 20% respectively of the market share. Based upon a study by a marketing research firm, the following facts were compiled. Bakery A retains 90% of its customers while gaining 5% of competitor B's customers and 10% of C's customers. Bakery B retains 85% its customers while gaining 5% of A's customers and 7% of c's customers. Bakery C retains 83% of its customers and gains 5% of A's customers and 10% of B's customers.
Determine the each firm's share be on January 1, and also the shares of each firm in next 4 year period.
Q5. A bank has one drive-in-counter. It is estimated that cars arrive according to Poisson distribution at the rate of 2 every 5 minutes and that there is enough space to accommodate a line of 10 cars. Other arriving cars can wait outside this space, if necessary. It takes 1-5 minutes on an average to serve a customer, but the service time actually varies according to an exponential distribution. You are required to find:
i) the proportion of time the facility remains idle;
ii) the expected number of customers waiting but currently not being served at a particular point of time;
iii) The expected time a customer spends in the system and,
iv) The probability that the waiting line will exceed the capacity of the space leading to the drive-in counter.
Part D -
Q1. The following are measurements of the Carbon content and Permeability index of 22 sinter mixtures.
|
Carbon content:
|
4.4
|
5.5
|
4.2
|
3.0
|
4.5
|
4.1
|
4.9
|
4.7
|
5.0
|
4.6
|
4.9
|
4.6
|
5.0
|
4.7
|
5.1
|
4.4
|
3.6
|
4.9
|
5.1
|
4.8
|
5.2
|
5.2
|
|
Permeability index:
|
12
|
14
|
18
|
35
|
23
|
13
|
19
|
22
|
20
|
16
|
29
|
16
|
12
|
18
|
21
|
27
|
27
|
21
|
13
|
18
|
17
|
11
|
Calculate Karlpearson correlation coefficient for the above data.
Q2. In a certain type of metal test specimen, the normal stress on a specimen is known to be functionally related to the shear resistance. The following is a set of coded experimental data on the two variables:
|
Normal Stress:
|
26.8
|
25.4
|
28.9
|
23.6
|
27.7
|
23.9
|
24.7
|
28.1
|
26.9
|
27.4
|
22.6
|
25.6
|
|
Shear resistance:
|
26.5
|
27.3
|
24.2
|
27.1
|
23.6
|
25.9
|
26.3
|
22.5
|
21.7
|
21.4
|
25.8
|
24.9
|
(i) Make a scatter plot.
(ii) Fit a straight line of the form y = a + bx.
(iii) Estimate the shear resistance for a normal stress of 24.5 kilograms per square centimeter.
Q3. A house wife buys three kinds of cereals; A, B an C. She never buys the same cereal on successive weeks. If she buys cereal A, then the next week she buys cereal B. Hower, if she buys either 13 or C, then the next week she is three times as likely to buy A as the other brand. Obtain the transition probability matrix and determine how often she would buy each of the cereals in the long run.
Q4. There are three dairies in a town say A, B and C. They supply all the milk consumed in the town. It is know by all the dairies that customers switch from dairy to dairy over time because of advertising, dissatisfaction with service and other reasons. All these dairies maintain records of the number of their customers and the dairy from which they obtained each new customer. Following table illustrates the flow of customers over an observation period of one month, say June.
|
Dairy
|
June 1 customers
|
Gains from
|
Losses to
|
July 1 customers
|
|
A
|
B
|
C
|
A
|
B
|
C
|
|
A
|
200
|
0
|
35
|
25
|
0
|
20
|
20
|
220
|
|
B
|
500
|
20
|
0
|
20
|
35
|
0
|
15
|
490
|
|
C
|
300
|
20
|
15
|
0
|
25
|
20
|
0
|
290
|
We assume the matrix of transition probabilities remains fairly s able and that the July 1 market shares are A = 22%, B = 49% and C = 29%. Managers of these dairies are willing to know,
i) Market shares to their dairies 1 august and 1st September and
ii) Their market shares in steady state.
Q5. A maintenance service facility has Poisson arrival rates, negative exponential service times, and operates on a first-come first-served queue discipline. Breakdowns occur on an average of three per day with a range of zero to eight. The maintenance crew can service on an average six machines per day with a range from zero to seven. Find the
i) utilization factor of the service facility,
ii) mean time in the system,
iii) mean number in the system in Break down repair,
iv) mean waiting time in the queue,
v) probability of finding two machines in the system.
Part E -
Q1. The following are measurements of the Carbon content and Permeability index of 22 sinter mixtures.
|
Carbon content:
|
4.4
|
5.5
|
4.2
|
3.0
|
4.5
|
4.1
|
4.9
|
4.7
|
5.0
|
4.6
|
4.9
|
4.6
|
5.0
|
4.7
|
5.1
|
4.4
|
3.6
|
4.9
|
5.1
|
4.8
|
5.2
|
5.2
|
|
Permeability index:
|
12
|
14
|
18
|
35
|
23
|
13
|
19
|
22
|
20
|
16
|
29
|
16
|
12
|
18
|
21
|
27
|
27
|
21
|
13
|
18
|
17
|
11
|
Calculate Karlpearson correlation coefficient for the above data.
Q2. An experiment was conducted in order to determine if cerebral blood flow in human beings can predicted from arterial oxygen. Fifteen patients were used in the study and the following data were observed.
|
Arterial Oxygen x:
|
603.40
|
582.50
|
556.20
|
594.60
|
558.90
|
575.20
|
580.10
|
451.20
|
404.00
|
484.00
|
452.40
|
448.40
|
334.80
|
320.30
|
350.30
|
|
Blood flow y:
|
84.33
|
87.80
|
82.20
|
78.21
|
78.44
|
80.01
|
83.53
|
79.46
|
75.22
|
76.58
|
77.90
|
78.80
|
80.67
|
86.60
|
78.20
|
Plot the data and also Estimate the Quadratic regression equation.
Q3. A store starts a week with at least 3 PCs. The demand per week is estimated at 0 with probability 0.15, 1 with probability 0.2. 2 with probability 0.35, 3 with probability 0.25 and 4 with probability 0.05. Untitled demand is backlogged. The store's policy is to place an order for delivery at the start or the following week whenever the inventory level drops below 3PCs. The new replinshment always brings the stock back to 5 PCs.
1) Express the situation as a Markov chain.
ii) Suppose that the week starts with 4 PCs. Determine the probability that an order will he placed at the end of two weeks.
Q4. Honey Inc had 35% of the local market for its cosmetics. while the two other manufactures of cosmetics Lace Lnc and shalon Inc have 40% and 25% shares respectively in the local market as on 1st April of this year. A study by a market research firm has disclosed the following:
Honey Inc. retains 86% of its customers, while it gains 4% and 6% of the customers from its two competitors. Lace and Shalon respectively, Lace, Inc. retains 90% of its customers, and gains 8% and 9% of customers respectively., from Honey and Shalon. Shalon retains 85% of its customers and gain 6% and 6% of customers from Lace and Honey, respectively.
i) Determine the share of each firm on April 1st next year.
ii) Determine the market share of each firm at equilibrium.
Q5. Arrivals at a telephone booth art considered to be Poisson, with an average time of 10 minutes between one arrival and the next. The length of a phone call assumed to be distributed exponentially with mean 3 minutes. Then
a) Find the probability that a person arriving at the booth will have to wait?
b) Find the average length of the queues that from time to time.