PS+1+work+book+solution(1)

1 5

Concept of Probability

Learning Outcome: The students should be able to understand the basic concept of probability, sample space, probability of events, counting rule; conditional probability; multiplication rule and Bayes theorem Exercise 1: (Example 1 – L1) Each message in a digital communication system is classified as to whether it is received within the time specified by the system design. If 3 messages are classified, what is an appropriate sample space for this experiment?

Exercise 2: (Example 2 – L1) A digital scale is used that provide weights to the nearest gram. Let event A: a weight exceeds 11 grams, B: a weight is less than or equal to 15 grams, C: a weight is greater than or equal to 8 grams and less than 12 grams. What is the sample space for this experiment? and find (a) A U B

(b) A’

(c) A ∩ B

(d) (A U C)’ (e) A ∩ B ∩ C (f) B’ ∩ C

1

Probability and Statistics Work Book

Exercise 3: (Example 3 – L1) Samples of building materials from three suppliers are classified for conformance to airquality specifications. The results from 100 samples are summarized as follows:

Supplier

R S T

Conforms Yes No 30 10 22 8 25 5

Let A denote the event that a sample is from supplier R, and B denote the event that a sample conforms to the specifications. If sample is selected at random, determine the following probabilities: (a) P(A) (b) P(B) (c) P(B’) (d) P(AUB) (e) P(A  B) (f) P(AUB’) (g) P ( A B ) (h) P ( B A)

2


Exercise 4: (Tutorial 1. (No. 1)) The compact discs from a certain supplier are analyzed for scratch and shock resistance. The results from 100 discs tested are summarized as follows: Scratch Resistance High Low 30 10 22 8 25 5

High Medium Low

Shock Resistance

Let A denote the event that a disc has high shock resistance, and B denote the event that a disc has high scratch resistance. If sample is selected at random, determine the following probabilities: (a) P(A) (b) P(B) (c) P(B’) (d) P(AUB) (e) P(A  B) (f) P(AUB’) (g) P ( A B ) (h) P ( B A)

SOLUTION: (a) (b) (c) (d) (e) (f) (g) (h)

P(A) = n(A)/N = 40/100 = 0.40 P(B) = n(B)/N = 77/100 = 0.77 P(B’) =n(B’)/N = 23/100 = 0.23 = 1 – P(B) P(AUB) = P(A)+P(B) – P(A B) = 0.4+0.77-0.3 = 0.87 P(A  B) = 0.3 P(AUB’) = P(A) +P(B’) - P(A B’) = 0.4 + 0.23 – 0.10 = 0.53 P ( A B ) = P(A B)/P(B) = 0.3/0.77 = 30/77 = 0.389 P ( B A) = P(A  B)/P(A) = 0.3/0.4 =0.75







Exercise 5:

3


The reaction times ( in minutes) of a reactor for two batches are measured in an experiment. a) Define the sample space of the experiment. b) Define event A where the reaction time of the first batch is less than 45 minutes and event B is the reaction time of the second batch is greater than 75 minutes. c) Find A U B, A ∩ B and A’ d) Verify whether events A and B are mutually exclusive.

Exercise 6: (Tutorial 1 (No:2) When a die is rolled and a coin is tossed, use a tree diagram to describe the set of possible outcomes and find the probability that the die shows an odd number and the coin shows a head. SOLUTION: S = {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T} A = { 1H, 3H, 5H} P(A) = 3/12 = 0.25 Exercise 7: (Tutorial 1 (No.3)) A bag contains 3 black and 4 while balls. Two balls are drawn at random one at a time without replacement. (i) (ii)

What is the probability that a second ball drawn is black? What is the conditional probability that first ball drawn is black if the second ball is known to be black?

SOLUTION: (i)

S={BB, BW, WB, WW} A={BB,WB} P(A)=P(BB)+P(WB)=6/42 + 12/42 =18/42 = 3/7

(ii)

C={BB,BW} P(C/A) = P(C∩A)/P(A) = P(BB)/P(A) = 1/3

4


Exercise 8: An oil-prospecting firm plans to drill two exploratory wells. Past evidence is used to assess the possible outcomes listed in the following table: Event

Description

Probability

A B C

Neither well produces oil or gas Exactly one well produces oil or gas Both wells produce oil or gas

0.80 0.18 0.02

Find and give description for

P( A  B ), P( B  C ) and P ( B ' )

Exercise 9: (Example 4 – L2) In a residential suburb, 60% of all households subscribe to the metro newspaper published in a nearby city, 80% subscribe to the local paper, and 50% of all households subscribe to both papers. Draw a Venn diagram for this problem. If a household is selected at random, what is the probability that it subscribes to a) at least one of the two newspapers b) exactly one of the two newspapers

5


Exercise 10: (Example 4 – L2) In a student organization election, we want to elect one president from five candidates, one vice president from six candidates, and one secretary from three candidates. How many possible outcomes?

Exercise 11: Suppose each student is assigned a 5 digit number. How many different numbers can be created?

Exercise 12: (Example 6 – L2) A chemical engineer wishes to conduct an experiment to determine how these four factors affect the quality of the coating. She is interested in comparing three charge levels, five density levels, four temperature levels, and three speed levels. How many experimental conditions are possible?

Exercise 13: (Tutorial 1 (No.4))

6


A menu has five appetizers, three soup, seven main course, six salad dressings and eight desserts. In how many ways can a) A full meal be chosen? b) A meal be chosen if either and appetizer or a soup is ordered, but not both? SOLUTION: a)

A full meal be chosen is 5 x 3 x 7 x 6 x 8 = 5040

b) A meal be chosen if either and appetizer or a soup is ordered, but not both is (5 + 3) x 7 x 6 x 8 = 2688

Exercise 14: (Example 7 – L3) Ten teaching assistants are available to grade a test of four questions. Wish to select a different assistant to grade each question (only one assistant per question). How many possible ways can the assistant be chosen for grading?

Exercise 15: (Example 8 – L3) Participant samples 8 products and is asked to pick the best, the second best, and the third best. How many possible ways?

Exercise 16: (Example 9 – L3) Suppose, that in the taste test, each participant samples eight products and is asked to select the three best product. What is the number of possible outcomes?

7


Exercise 17: A contractor has 8 suppliers from which to purchase electrical supplies. He will select 3 of these at random and ask each supplier to submit a project bid. In how many ways can the selection of bidders be made?

Exercise 18: (Example 10 – L2) Twenty players compete in a tournament. In how may ways can a) rankings be assigned to the top five competitors? b) the best five competitors be randomly chosen?

Exercise 19: (Example 11 – L3) Three balls are selected at random without replacement from the jar which contains 3 black, 2 red and 3 green balls. Find the probability that one ball is red and two are black.

Exercise 20: A university warehouse has received shipment of 25 printers, of which 10 are laser printers and 15 are inkjet models. If 6 of these 25 are selected at random by a technician, what is the probability that exactly 3 of those selected are laser printers?

Exercise 21:

8


There are 17 broken light bulbs in a box of 100 light bulbs. A random sample of 3 light bulbs is chosen without replacement. a) How many ways are there to choose the sample? b) How many samples contain no broken light bulbs? c) What is the probability that the sample contains no broken light bulbs? d) How many ways to choose a sample that contains exactly 1 broken light bulb? e) What is the probability that the sample contains no more than 1 broken light bulb?

Exercise 22: (Tutorial 1 (No. 5)) An agricultural research establishment grows vegetables and grades each one as either good or bad for taste, good or bad for its size, and good or bad for its appearance. Overall, 78% of the vegetables have a good taste. However, only 69% of the vegetables have both a good taste and a good size. Also, 5% of the vegetables have a good taste and a good appearance, but a bad size. Finally, 84% of the vegetables have either a good size or a good appearance. a) if a vegetable has a good taste, what is the probability that it also has a good size? b) if a vegetable has a bad size and a bad appearance, what is the probability that it has a good taste? SOLUTION: Let event T : good taste , event S: good size and event A: good appearance

a)

P (T )  0.78 ,

P (T  S )  0.69 ,

P (T  S ' A)  0.05 , P ( S  A)  0.84 P (T  S ) 0.69 P( S | T )    0.885 P (T ) 0.78

b) P ( S ' A' )  0.16 ,

P (T  S ' )  0.09 ,

P (T  S ' A' )  0.04 , P (T  S ' A' ) 0.04 P (T | S ' A' )    0.25 P ( S ' A' ) 0.16 Exercise 23: (Example 12 – L4)

9


A local library displays three types of books entitled “Science” (S), “Arts” (A), and “Novels” (N). Reading habits of randomly selected reader with respect to these types of books are Read regularly Probability

S 0.14

A 0.23

N 0.37

S∩A S∩N 0.08 0.09

A∩N 0.13

S∩A∩N 0.05

Find the following probabilities and interpret a) P( S | A ) b) P( S | A U N ) c) P( S | reads at least one ) d) P( S U A | N)

Exercise 24: (Tutorial 1 (No.6)) A batch of 500 containers for frozen orange juice contains 5 that are defective. Two are selected at random, without replacement, from the batch. Let A and B denote that the first and second selected is defective respectively a) Are A and B independent events? b) If the sampling were done with replacement, would A and B be independent? SOLUTION: Using tree diagram: a)

P (B|A) = 4/499

P(B) = P(B|A)P(A) + P(B|A’)P(A’) = 5/500

Because P (B|A) not equal to P(B) , then A and B are not independent events b)

A and B are independent events because P (B|A) = P(B) = 5/500

Exercise 25: (Example 13 – L4)

10


Everyday (Mon to Fri) a batch of components sent by a first supplier arrives at certain inspection facility. Two days a week, a batch also arrives from a second supplier. Eighty percent of all batches from supplier 1 pass inspection, and 90% batches of supplier 2 pass inspection. On a randomly selected day, what is the probability that two batches pass inspection?

Exercise 26: The probability is 1% that an electrical connector that is kept dry fails during the warranty period of a portable computer. If the connector is ever wet, the probability of a failure during the warranty period is 5%. If 90% of the connectors are kept dry and 10% are wet, what proportion of connectors fail during the warranty period?

Exercise 27: (Example 14 – L5) Computer keyboard failures are due to faulty electrical connects (12%) or mechanical defects (88%). Mechanical defects are related to loose keys (27%) or improper assembly (73%). Electrical connect defects are caused by defective wires (35%), improper connections (13%) or poorly welded wires (52%).


11


During a space shot, the primary computer system is backed up by two secondary systems. They operate independently of one another, and each is 90% reliable. What is the probability that all three systems will be operable at the time of the launch?

Exercise 29: A store stocks light bulbs from three suppliers. Suppliers A, B, and C supply 10%, 20%, and 70% of the bulbs respectively. It has been determined that company A’s bulbs are 1% defective while company B’s are 3% defective and company C’s are 4% defective. If a bulb is selected at random and found to be defective, what is the probability that it came from supplier B?

Exercise 30: (Example 16 – L5) A particular city has three airports. Airport A handles 50% of all airline traffic, while airports B and C handle 30% and 20%, respectively. The rates of losing a baggage in airport A, B and C are 0.3, 0.15 and 0.14 respectively. If a passenger arrives in the city and losses a baggage, what is the probability that the passenger arrives at airport A?


12


A company rated 75% of its employees as satisfactory and 25% unsatisfactory. Of the satisfactory ones 80% had experience, of the unsatisfactory only 40%. If a person with experience is hired, what is the probability that (s)he will be satisfactory?

Exercise 32: In a certain assembly plant, three machines, B1, B2, B3, make 30%, 45% and 25%, respectively, of the products. It is known from past experience that 2%,3% and 2% of the products made by each machine, respectively, are defective. Now, suppose that a finished product is randomly selected. What is the probability that it is defective?

Exercise 33: (Tutorial 1 (No. 7)) Three machines A, B and C produce identical items of their respective output 5%, 4% and 3% of the items are faulty. On a certain day A has produced 25%, B has produced 30% and C has produced 45% of the total output. An item selected at random is found to be faulty. What are the chance that it was produced by C? SOLUTION: Let F be the event that the item is faulty. (i)

P(F) = P(FA)+P(FB)+P(FC)=(0.25)(0.05)+(0.30)(0.04)+(0.45)(0.03) = 0.038

(ii)

P(C |F) =P(FC)/P(F) = (0.45)(0.03)/0.038 = 0.355

Exercise 34: (Tutorial 1 (No. 8))

13


Suppose that a test for Influenza A, H1N1 disease has a very high success rate: if a tested patient has the disease, the test accurately reports this, a ’positive’, 99% of the time, and if a tested patient does not have the disease, the test accurately reports that, a ’negative’, 95% of the time. Suppose also, however, that only 0.1% of the population have that disease. (i) (ii) (iii)

What is the probability that the test returns a positive result? If the patient has a positive, what is the probability that he has the disease? What is the probability of a false positive?

SOLUTION: Let D be the event that the patient has the disease, and E be the event that the test returns a positive result. (i)

P(E) = P(DE) + P(DCE) = (0.001)(0.99) + (0.999)(0.05) = 0.05094

(ii)

The probability that the patient has the disease given that he has a positive test actually is the probability of a true positive, that is P(D |E) = P(DE)/P(E) = (0.001)(0.99)/0.05094 = 0.019 –

(iii)

P(False Positive) = 1 - P(D |E) = 1 – 0.019 = 0.981

Exercise 35: An insurance company charges younger drivers a higher premium than it does older drivers because younger drivers as a group tend to have more accidents. The company has 3 age groups: Group A includes those less than 25 years old, have a 22% of all its policyholders. Group B includes those 25-39 years old, have a 43% of all its policyholders, Group C includes those 40 years old and older, have 35% of all its policyholders. Company records show that in any given one-year period, 11% of its Group A policyholders have an accident. The percentages for groups B and C are 3% and 2%, respectively. (a)

What is the probability that the company’s policyholders are expected to have an accident during the next 12 months?

(b)

Suppose Mr. Chong has just had a car accident. If he is one of the company’s policyholders, what is the probability that he is under 25?

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2 1

Descriptive Statistics

Learning Outcome: The students should be able to understand the basic concept of statistics, descriptive statistics, measures of tendency, measures of dispersions and data display. Exercise 1: (Example 1 – L1)

Find the mean, median and mode for the following observations: 6.5

7.8

4.6

3.7

6.5 9.2

12.1

6.5

3.7

10.8

Exercise 2: (Tutorial 2. (No. 1)) Find the mean, median and mode for the following observations : 2.3 3.6

2.6

2.8

3.2

3.6

4.3

5.2

6.9

2.8 3.6

SOLUTION: Arrange the observations in increasing order: n = 11 Mean = 40.9/11 = 3.718 Median = 3.6 Mode = 3.6 Exercise 3: (Example 2 – L1) Seven oxide thickness measurements of wafers are studied to assess quality in a semiconductor manufacturing process. The data (in angstroms) are: 1264, 1280, 1301, 1300, 1292, 1307, and 1275. Calculate the sample average, variance and standard deviation.

15


Exercise 4: (Tutorial 2. (No. 2)) The following data are direct solar intensity measurements (watts/m2) on different days at a location in southern Spain: 562, 869, 708, 775, 775, 704, 809, 856, 655, 806, 878, 909, 918, 558, 768, 870, 918, 940, 946, 661, 820, 898, 935, 952, 957, 693, 835, 905, 939, 955, 960, 498, 653, 730, 753. Calculate the sample mean, variance and sample standard deviation. SOLUTION: Sample average:

Sample variance:

Sample standard deviation:

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Exercise 5: (Tutorial 2. (No.3)) Find the mean, variance and standard deviation of the following samples of marks for the engineering mathematics 1 final examination. 84.9 75.0 69.3 59.5 48.2 38.3

81.9 73.8 68.6 58.3 48.0 37.4

80.8 72.7 67.5 58.5 47.8 36.8

79.4 72.6 66.8 57.6 46.5 36.5

78.2 71.4 65.2 56.9 45.9 35.6

76.5 70.9 64.4 55.2 44.6 34.9

SOLUTION: Mean = 2166.4 / 36 = 60.177 Variance = 234.177 Standard deviation = 15.302 Exercise 6: Find the mean, variance and standard deviation of the following samples of marks for the engineering drawing course. 98.4 89.6 79.7 69.8 59.2 59.6 39.8

98.1 88.7 78.2 68.6 58.3 48.0 37.4

98.0 87.3 77.4 67.5 58.5 47.8 36.8

97.8 86.8 77.4 66.8 57.6 46.5 36.5

96.4 85.7 76.8 65.2 56.9 45.9 35.6

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95.2 94.3 92.6 91.8 84.2. 83.7 82.8 80.5 75.9 74.2 73.9 72.6 64.4 63.7 62.8 61.4 55.2 54.7 53.9 52.9 44.6 43.8 42.7 41.8 34.9 33.2 33.8 32.7

90.5 80.8 71.4 60.7 51.2 40.6 31.6


Exercise 7: (Example 3 – L2) The shear strengths of 100 spot welds in a titanium alloy follow. Construct a stem-and-leaf diagram for the weld strength data and comment on any important features that you notice. 5408 5431 5475 5442 5376 5388 5459 5422 5416 5435 5420 5429 5401 5446 5487 5416 5382 5357 5388 5457 5407 5469 5416 5377 5454 5375 5409 5459 5445 5429 5463 5408 5481 5453 5422 5354 5421 5406 5444 5466 5399 5391 5477 5447 5329 5473 5423 5441 5412 5384 5445 5436 5454 5453 5428 5418 5465 5427 5421 5396 5381 5425 5388 5388 5378 5481 5387 5440 5482 5406 5401 5411 5399 5431 5440 5413 5406 5342 5452 5420 5458 5485 5431 5416 5431 5390 5399 5435 5387 5462 5383 5401 5407 5385 5440 5422 5448 5366 5430 5418 (i) (ii)

Construct a stem-and-leaf display for these data. Find the median, the quartiles, and the 5th and 95th percentiles.

18


Exercise 8: (Tutorial 2. (No.4)) The data that follow represent the yield on 90 consecutive batches of ceramic substrate to which a metal coating has been applied by a vapor-deposition process.

(i) (ii) (iii)

94.1

87.3

94.1

92.4

84.6

85.4

93.2

84.1

92.1

90.6

83.6

86.6

90.6

90.1

96.4

89.1

85.4

91.7

91.4

95.2

88.2

88.8

89.7

87.5

88.2

86.1

86.4

86.4

87.6

84.2

86.1

94.3

85.0

85.1

85.1

85.1

95.1

93.2

84.9

84.0

89.6

90.5

90.0

86.7

78.3

93.7

90.0

95.6

92.4

83.0

89.6

87.7

90.1

88.3

87.3

95.3

90.3

90.6

94.3

84.1

86.6

94.1

93.1

89.4

97.3

83.7

91.2

97.8

94.6

88.6

96.8

82.9

86.1

93.1

96.3

84.1

94.4

87.3

90.4

86.4

94.7

82.6

96.1

86.4

89.1

87.6

91.1

83.1

98.0

84.5

Construct a cumulative frequency plot and histogram for the yield Construct a stem-and-leaf display for these data. Find the median, the quartiles, and the 5th and 95th percentiles for the yield

SOLUTION: (i)

19


20


(ii)

(iii)

Stem-and-leaf display for yield: unit = 78|3 represents 78.3

F

S

LEAF

1 2 4 8 6 10 7 5 6 10 4 3 5 8 4 4 2 1 90

78 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

3 6 0 0 0 1 3 2 1 0 1 1 1 1 1 1 3 0

9 1 1 1 1 3 2 1 0 2 4 1 1 2 3 8

6 1 1 1 3 3 4 1 4 4 2 1 3 4

7 1 1 4 5 6 6 1 7 2 3 6 8

2 4 4 6 8 6 3

7 3

5 4 4 6

6

9

4 7

6

6

7

7 4

5

6

6

6

4

6

7

Median = 89.25, 1st Q = 86.1, 3rd Q = 93.1, 5th% = 83.325, 95th% = 96.355

21


Exercise 9: The average age of the football players on each team of the premier league as follows.

(i) (ii) (iii)

29.4

29.8

29.4

31.8

32.7

34.0

28.5

27.9

30.9

29.3

28.8

28.6

29.1

31.0

30.7

30.3

29.7

31.0

28.4

28.9

27.7

28.7

30.5

29.8

26.6

27.9

27.9

29.9

29.3

28.1

Construct a cumulative frequency plot and histogram for the yield Construct a stem-and-leaf display for these data. Find the median, the quartiles, and the 5th and 95th percentiles for the yield

Exercise 10: (Example 4 – L2) The following “ cold start ignition time” of an automobile engine obtained for a test vehicle are as follows: 1.75 1.92 2.62 2.35 3.09 3.15 2.53 1.91 a) b)

Calculate the sample median, the quartiles and the IQR Construct a box plot of the data.

22


Exercise 11: (Tutorial 2. (No. 5)) The following data are the joint temperatures of the O-rings (°F) for each test firing or actual launch of the space shuttle rocket motor (from Presidential Commission on the Space Shuttle Challenger Accident, Vol. 1, pp. 129–131): 84, 49, 61, 40, 83, 67, 45, 66, 70, 69, 80, 58, 68, 60, 67, 72, 73, 70, 57, 63, 70, 78, 52, 67, 53, 67, 75, 61, 70, 81, 76, 79, 75, 76, 58, 31. (i) (i) (ii) (iii) (iv) (v)

Compute the sample mean and sample standard deviation; Find the upper and lower quartiles of temperature; Find the upper and lower quartiles of temperature; Find the median; Set aside the smallest observation (31°F) and recomputed the quantities in parts (a), (b), and (c). Comment on your findings. How “different” are the other temperatures from this smallest value?; and Construct a box plot of the data and comment on the possible presence of outliers.

Solution: (i) (ii)

Mean: 65.86, Standard Deviation: 12.16 Q1: 58.5, Q3: 75

(iii) Median: 67.5 (iv)

(v)

Mean: 66.86, Standard Deviation: 10.74, Q1: 60, Q3: 75, Median: 68 The mean has increased while the sample standard deviation has decreased. The lower quartile has increased while the upper quartile has remained unchanged. The median has increased slightly due to the removal of the data point. The smallest value appears quite different than the other temperature values. Using the entire data set, the box plot is

23


The value of 31 appears to be one possible outlier.

Exercise 12: Ipoh Pantai Hospital compiles data on the length of stay by patients in short-term hospitals. A random sample of 21 patients yielded the following data on length of stay, in days. 3 4 5 (i) (ii) (iii) (iv) (v)

6 15 4 12 10 13

7 3 55 1 18 9 6 12 7 1 23 9

Compute the sample mean and sample standard deviation; Find the upper and lower quartiles of temperature; Find the upper and lower quartiles of temperature; Find the median; Construct a box plot of the data and comment on the possible presence of outliers.

24

3 2


Random Variables and Discrete Probability Distribution

Exercise 1: (Tutorial 3. (No.1)) Identify each of the random variables as continuous or discrete random variable. (a) The number of atoms in a molecule (b) The number of ducks on a pond (c) The home team score in a basketball game (d) The voltage on a power line (e) A score on the mathematic final exam (f) The volume of water in the tank (g) The number of fish caught by a fishing boat (h) The number of traffic accidents in Tronoh (i) The number of doughnuts left in the pantry (j) The weight of an engineering student in UTP Solution: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

Discrete Discrete Discrete Continuous Continuous Continuous Discrete Discrete Discrete Continuous

25


Exercise 2: (Example 1 – L1) Let, x

(i) (ii) (iii) (iv)

0

1

2

0.3 5

0. 5

k

Find the value of k that result in a valid probability distribution. Find the expected number of x and the standard deviation. Find the variance and the standard deviation of x What is the probability that x greater than or equal to 1?

Exercise 3: (Tutorial 3. No. 2)) At UTP, the business students run an investment club. Each semester they create investment portfolios in multiples of RM1, 000 each. Records from the past several years show the following probabilities of profits (rounded to the nearest RM50). In the table below, x = profit per RM1, 000 and P(x) is the probability of earning that profit. x

(a) (b) (c) (d) (e)

0

50

10 0

15 0

200

0.1 5

0.3 5

k

0.2 0.0 5

Determine the value of k that results in a valid probability distribution. The profit per RM1, 000 is a random variable. Is it discrete or continuous? Explain. Find the expected value of the profit in a $1,000 portfolio. Find the standard deviation of the profit. What is the probability of a profit of $150 or more in a RM1, 000 portfolio?

Solution:

(a) (b) (c)

k = 1 – (0.15+0.35+0.2+0.05) = 0.25; Discrete since the profit is in term of integer value; E(X) = (0)(0.15)+(50)(0.35)+(100)(0.25)+(150)(0.2)+(200)(0.05) = 82.5

26


(d) e)

Sd(X) =  V ( X )  3068.75  55.3963 P ( X  150)  1  P ( X  150)  1  P ( X  200)  1  0.05  0.95

Exercise 4: (Example 2 – L1) Let X denote the number of bars of service on your cell phone whenever you are at an intersection with the following probabilities: x

0

1

2

3

4

5

0.1 0.15 0.25 0.25 0.15 0.1 Determine the following: (d) F(x) (e) Mean and variance (f) P(X < 2) (g) P(X < 3.5)

Exercise 5: (Tutorial 3. (No. 3)) A local cab company is interested in the number of pieces of luggage a cab carries on a taxi run. A random sample of 260 taxi runs gave the following information. x = number of pieces of luggage and f is the frequency with which taxi runs carried x pieces of luggage. x: f: (a) (b) (c) (d) (e)

0 42

1 51

2 63

3 4 38 19

5 16

6 12

7 10

8 6

9 2

10 1

Find the probability distribution for x. Make a histogram of the probability distribution. Estimate the probability that a taxi run will have from 0 to 4 pieces of luggage (including 0 and 4). What is the expected value of x? What is the standard deviation of x.

Solution: (a)

X: f:

0 42

1 51

2 63

3 38

27

4 19

5 16

6 12

7 10

8 6

9 2

10 1


P(X):

0.16 0.20 0.24 0.15 0.07 0.06 0.05 0.04 0.02 0.01 0.00

(b)

(c)

P (0  X  4)  P ( X  0)  P ( X  1)  P ( X  2)  P ( X  3)  P ( X  4) = 0.16 + 0.20 + 0.24 + 0.15 + 0.07 = 0.82

(d) (e)

E(X)=2.55 SD(X) = 4.66

Exercise 6: (Example 3 – L2) A Professor estimates the probability that he will receive at least one telephone call at home during the hours of 5pm to 7pm on a weekday to be 2/3. Use the formulas for computing binomial probabilities to answer the following questions: (a) What is the probability that he will receive at least one call on all five of the next five weekday nights? (b) What is the probability that he will not receive a call on any of the next five weekday nights? (c) What is the probability that he will receive a call on at least four of the next five weekday nights?

28


Exercise 7: (Tutorial 3. (No.4)) The probability of successfully landing a plane using a flight simulator is given as 0.80. Nine randomly and independently chosen student pilots are asked to try to fly the plane using the simulator. (a) (b) (c)

What is the probability that all the student pilots successfully land the plane using the simulator? What is the probability that none of the student pilots successfully lands the plane using the simulator? What is the probability that exactly eight of the student pilots successfully land the plane using the simulator?

Solution: (a)

The probability that all the student pilots successfully land the plane using the simulator is

(b)

The probability that none of the student pilots successfully lands the plane using the simulator is

(c)

The probability that exactly eight of the student pilots successfully land the plane using the simulator is

Exercise 8: (Quiz 2) Suppose X has a Poisson distribution a mean of 5 Determine the following. (a) P(X = 0);

29


(b) P(X = 5); (c) P(X < 3); and P ( X  4) . (d) Exercise 9: (Example 4 – L2) At the Mc Donald drive-thru window of food establishment, it was found that during slower periods of the day, vehicles visited at the rate of 12 per hour. Determine the probability that (a) no vehicles visiting the drive-thru within a ten-minute interval during one of these slow periods; (b) only 2 vehicles visiting the drive-thru within a ten-minute interval during one of these slow periods; and (c) at least three vehicles visiting the drive-thru within a ten-minute interval during one of these slow periods.

Exercise 10: (Tutorial 3. (No.5)) The number of cracks in a section of PLUS highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per kilometer. Determine the probability that (a) there are no cracks at all in 2km of highway; (b) at least one crack in 500meter of highway; and (c) there are exactly 3 cracks in 0.5km of highway. Solution:

40 e 4  0.0183 0!

(a)

P ( X  0) 

(b)

P ( X  1)  1  P ( X  0)  1 

(c)

P( X  3) 

10 e 1  0.6321 0!

13 e 1  0.0613 3!

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4 3


Continuous Probability Distribution

Exercise 1: (Example 1 – L1) Suppose that X is a continuous random variable having the probability density function

a) b) c)

 1.5x 2 for 1  x  1

f (x )  

 0,

elsewhere

Show that f(x) is a probability density function Find P(-0.5
Exercise 2: (Example 2 – L1) Let X be a continuous random variable with pdf given by

Find (i) (ii) (iii) (iv)

 kx ,  1  x  2 f ( x)    0 , elsewhere the value of constant k; P(X < 1); the mean of X; and the standard deviation of X.

31


Exercise 3: (Tutorial 4. (No.1)) Let X be a continuous random variable with pdf given by

 kxe2x , x  0

f ( x)  

,x  0

0

Find (i) (ii) (iii) (iv) (v)

the value of constant k; P(X > 1); P(0 < X < 2) the mean of X; and the variance of X.

Solution: 

(i)  kxe  2 x dx   k 0

1 xe  2 x 2



0



 k 0

1 2 x 1 e dx   e  2 x 2 4



k  0

1 k 1 k  4 4

(ii)

P ( X  1) 





2 x 2 x  4 xe dx   2 xe

1

1



  2e  2 x dx  2e  2  e  2  3e  2  0.406 1

(iii) 2

P (0  X  2)   4 xe  2 x dx   2 xe  2 x 0

 5e (iv)

4

2 0

2

  2e  2 x dx  4e  4  [e  4  1] 0

 1  0.9084

mean( X )  E ( X ) 



 4x

2

e  2 x dx  1

0

32


E( X 2 )  (v)



 4x

3

e  2 x dx 

0

 Var ( X ) 

3 2

3 1  12  2 2

Exercise 4: (Example 3 – L1) Let X be a continuous random variable with pdf given by

 kx 2

f ( x)   Find (i) (ii) (iii) (iv) (v)

0

,0  x  1 , elsewhere

the value of constant k; the cdf, F(x); P(0.25 < X < 0.5) the mean of X; and the variance of X.

Exercise 5: Find the cumulative probability distribution of X given that the density function is

Find (i) (ii) (iii) (iv) (v)

 k (1  x4 ), for 0  x  1 f ( x)    0, elsewhere

the value of constant k; the cdf, F(x); P(0.25 < X < 0.5) the mean of X; and the variance of X.

33


Exercise 6: (Tutorial 4. (No.2)) Suppose a random variable, X has a uniform distribution with a = 5 and b = 9. Find a. the cdf, F(x) b. P(5.5 < X < 8). c. P(X < 7). d. the mean of X; and e. the standard deviation of X. Solution:

1  f ( x)   4  0

Pdf of X is



  

(b)

, elsewhere

x

  0dx  0

(a)

,5  x  9

5

for x  5

x

1 1 F ( x)    0dx   dx  x for 5  x  9  - 5 4 4  0 91    0dx   dx   0dx  1 for x  9  - 5 4 9 ½

34


(c) (d) (e)

½ Mean(X)=7 SD(X)=1.154

Exercise 7: (Example 4 – L2) Let X be an exponential random variable with λ = 0.5. Calculate the following probabilities: a. P(X < 5) b. P(x > 6) c. P(5 < x < 6) d. the probability that X is at most 6

Exercise 8: (Example 5 – L2) The lifetime of a certain electronic component is known to be exponentially distributed with a mean lifetime of 100 hours. What is the probability that (i) the lifetime of the component is more than 100hours? (ii) the lifetime of the component is between 50 to 100hours? (iii) a component will fail before 50hours?

35


Exercise 9: (Tutorial 4. (No. 3)) The time between telephone calls to ASTRO, a cable television payment processing center follows an exponential distribution with a mean of 1.5 minutes. What is the probability that the time between the next two calls (i). at least 45 seconds? (ii). will be between 50 to 100 seconds?; and (iii) at most 150 seconds? Solution:

 2 2 / 3 x  e ,x0 pdf for X is f ( x)   3  0 , x  0 (i) (ii) (iii)



45 2 2 / 3x )  e dx   e  2 / 3 x 60 3 45 / 60 P(50/60 < X < 100/60) =0.2446 P(X < 150/60) = 0.8111 P( X 

 45 / 60

 0.6065

Exercise 10: (Example 6 – L2) The braking distances on a dry surface for a certain vehicle are normally distributed with mean 160m and standard deviation 5m. What is (i) the chance it takes more than 155m to stop?

36


(ii) (iii)

the probability it takes between 150m to 170m to stop? the braking distance if 90% chance that the vehicle will stop?

Exercise 11: (Tutorial 4. (No. 4)) An average LCD Projector bulb manufactured by the ABC Corporation lasts 300 days with variance of 2500days. By assuming that the bulb life is normally distributed, what is the probability that the bulb will last (i) at most 365 days? (ii) between 250days and 350days? (iii) at least 400days? Solution:

365  300 )  P ( Z  1.3)   (1.3)  0.9032 50

(i)

P ( X  365)  P ( Z 

(ii) (iii)

P(250 < X < 350)=P(-1 < Z < 1) = 0.6827 P(X > 400)=P(Z > 2)=1-0.02275

Exercise 12: The line width of a tool used for semiconductor manufacturing is assumed to be normally distributed with a mean of 0.5 micrometer and a standard deviation of 0.05 micrometer. (i) What is the probability that a line width is greater than 0.62 micrometer? (ii) What is the probability that a line width is between 0.47 and 0.63 micrometer? (iii) The line width of 90% of samples is below what value?

37


38

PS+1+work+book+solution(1)

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