Miss
Shumailas
Handouts
Features: All her Handouts
CamScanner
CamScanner
CamScanner
CamScanner
CamScanner
CamScanner
CamScanner
CamScanner
NED UNIVERSITY OF ENGINEERING & TECHNOLOGY
SE (EE) SPRING SEMESTER 2024
CHAPTER 4 & 5
DISCRETE RANDOM VARIABLE/DISCRETE PROBABILITY DISTRIBUTION
Handout #2 P&S (MT-331)
DISCRETE RANDOM VARIABLE
1.
The probability distribution of the discrete random variable X is Find the mean of X.
2.
Roulette wheel is divided in to 6 sectors of unequal area marked with the no’s 1, 2, 3, 4, 5, & 6. The wheel
is spun & X is the random variable the number on which the wheel stops. The probability of X is as follows:
x
1
2
3
4
5
6
P(X=x)
1/16 3/16 k
1/4 3/16 1/16
Find the value of ‘k’ and find (i) E(3x-5)
(ii) E (6 x  6 x  10)
2
3.
Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the
probabilities are 0.22, 0.36, 0.28, and 0.14, respectively, that she will be able to sell it for a profit of $250,
sell it for a profit of $150, break even, or sell it for a loss of $150.What is her expected profit?
4.
An attendant at a car wash is paid according to the number of cars that pass through. Suppose the
probabilities are 1/12, 1/12, 1/4, 1/4, 1/6, and 1/6, respectively, that the attendant receives $7, $9, $11,
$13, $15, or $17 between 4:00 P.M. and 5:00 P.M. on any sunny Friday. Find the attendant’s expected
earnings for this particular period.
5.
The number of messages sent per hour over a computer network has the following distribution:
Determine the mean and standard deviation of the number of messages sent per hour.
6.
The monthly demand for transistors is known to have the following probability distribution.
Determine the expected demand for transistors. Also obtain the variance?
Shumaila Usman (Assistant professor)
(Mathematics Department)
NED UNIVERSITY OF ENGINEERING & TECHNOLOGY
SE (EE) SPRING SEMESTER 2024
CHAPTER 4 & 5
DISCRETE RANDOM VARIABLE/DISCRETE PROBABILITY DISTRIBUTION
Handout #2 P&S (MT-331)
BINOMIAL & MULTINOMIAL DISTRIBUTION
1
Twelve people are given two identical speakers, which they are asked to listen to for differences, if any. Suppose that these
People answer simply by guessing. Find the probability that three people claim to have heard a difference between the two
speakers.
2
In a certain city district, the need for money to buy drugs is stated as the reason for 75% of all thefts. Find the probability that
among the next 5 theft cases reported in this district,
(a) Exactly 2 resulted from the need for money to buy drugs?
(b) At most 3 resulted from the need for money to buy drugs?
3
2% of the batteries produced by a factory are not satisfactory. In a sample of 12 batteries chosen at random, find the
probability that:
a) Exactly 3 batteries are not satisfactory?
b) Less than 2 batteries are not satisfactory?
4
According to a survey by the Administrative Management Society, one-half of U.S. companies give employees 4 weeks of
vacation after they have been with the company for 15 years. Find the probability that among 6 companies surveyed at
random, the number that give employees 4 weeks of vacation after 15 years of employment is
(a) anywhere from 2 to 5;
(b) Fewer than 3.
5
One prominent physician claims that 70% of those with lung cancer are chain smokers. If his assertion is correct,
(a) find the probability that of 10 such patients recently admitted to a hospital, fewer than half are chain smokers;
(b) Find the probability that of 20 such patients recently admitted to a hospital, fewer than half are chain smokers.
6
In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the trucks fail to complete the test run
without a blowout. Of the next 15 trucks tested, find the probability that
(a) from 3 to 6 have blowouts;
(b) fewer than 4 have blowouts;
(c) More than 5 have blowouts.
7
The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next
7 patients having this operation survive?
8
A traffic control engineer reports that 75% of the vehicles passing through a checkpoint are from within the state. What is the
probability that fewer than 4 of the next 9 vehicles are from out of state?
9
It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the
probability that
(a) none contracts the disease.
(b) fewer than 2 contract the disease.
(c) More than 3 contract the disease.
10
Suppose that airplane engines operate independently and fail with probability equal to 0.4. Assuming that a plane makes a
safe flight if at least one-half of its engines run, determine whether a 4-engine plane or a 2- engine plane has the higher
probability for a successful flight.
Shumaila Usman (Assistant professor)
(Mathematics Department)
NED UNIVERSITY OF ENGINEERING & TECHNOLOGY
SE (EE) SPRING SEMESTER 2024
CHAPTER 4 & 5
DISCRETE RANDOM VARIABLE/DISCRETE PROBABILITY DISTRIBUTION
Handout #2 P&S (MT-331)
11
A multiple choice test contains 25 questions, each with four answers. Assume a student just guesses on each question.
(a) What is the probability that the student answers more than 20 questions correctly?
(b) What is the probability the student answers less than 5 questions correctly?
12
The probabilities are 0.4, 0.2, 0.3, and 0.1, respectively, that a delegate to a certain convention arrived by air, bus,
automobile, or train. What is the probability that among 9 delegates randomly selected at this convention, 3 arrived by air, 3
arrived by bus, 1 arrived by automobile, and 2 arrived by train?
13
A company generally purchases large lots of a certain kind of electronic device. A method is used that rejects a lot if 2 or more
defective units are found in a random sample of 25 units.
(a) What is the probability of rejecting a lot that is 1% defective?
(b) What is the probability of accepting a lot that is 5% defective?
14
The probability of a bomb hitting a target is 0.20. Two bombs are enough to destroy a bridge. If 6 bombs are aimed at the
bridge, find the probability that the bridge will be destroyed.
15
A safety engineer claims that only 40% of all workers wear safety helmets when they eat lunch at the workplace. Assuming
that this claim is right, find the probability that 4 of 6 workers randomly chosen will be wearing their helmets while having
lunch at the workplace.
NEGATIVE BINOMIAL DISTRIBUTION , GEOMETRIC DISTRIBUTION & POISSON DISTRIBUTION
16
The probability that a person living in a certain city owns a dog is estimated to be 0.3. Find the probability that the tenth
person randomly interviewed in that city is the fifth one to own a dog.
17
Find the probability that a person flipping a coin gets
(a) the third head on the seventh flip;
(b) The first head on the fourth flip.
18
Three people toss a fair coin and the odd one pays for coffee. If the coins all turn up the same, they are tossed again. Find
the probability that fewer than 4 tosses are needed.
19
A scientist inoculates mice, one at a time, with a disease germ until he finds 2 that have contracted the disease. If the
probability of contracting the disease is 1/6, what is the probability that 8 mice are required?
20
The probability that a student pilot passes the written test for a private pilot’s license is 0.7. Find the probability that a given
student will pass the test
(a) on the third try.
(b) Before the fourth try.
21
Suppose the probability that any given person will believe a tale about the transgressions of a famous actress is 0.8. What is
the probability that
(a) The sixth person to hear this tale is the fourth one to believe it?
(b) The third person to hear this tale is the first one to believe it?
Shumaila Usman (Assistant professor)
(Mathematics Department)
NED UNIVERSITY OF ENGINEERING & TECHNOLOGY
SE (EE) SPRING SEMESTER 2024
CHAPTER 4 & 5
DISCRETE RANDOM VARIABLE/DISCRETE PROBABILITY DISTRIBUTION
Handout #2 P&S (MT-331)
22
An inventory study determines that, on average, demands for a particular item at a warehouse are made 5 times per day.
What is the probability that on a given day this item is requested
(a) More than 5 times?
(b) Not at all?
23
A certain area of the eastern United States is, on average, hit by 6 hurricanes a year. Find the probability that in a given year
that area will be hit by
(a) fewer than 4 hurricanes.
(b) Anywhere from 6 to 8 hurricanes.
24
On average, 3 traffic accidents per month occur at a certain intersection. What is the probability that in any given month at
this intersection
(a) Exactly 5 accidents will occur?
(b) Fewer than 3 accidents will occur?
(c) At least 2 accidents will occur?
25
A restaurant chef prepares a tossed salad containing, on average, 5 vegetables. Find the probability that the salad contains
more than 5 vegetables
(a) on a given day
(b) on 3 of the next 4 days
(c) For the first time in April on April 5.
26
The probability of a successful optical alignment in the assembly of an optical data storage product is 0.8. Assume the trials
are independent.
(a) What is the probability that the first successful alignment requires exactly four trials?
(b) What is the probability that the first successful alignment requires at most four trials?
(c) What is the probability that the first successful alignment requires at least four trials?
27
The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a
Poisson distribution with a mean of two cracks per mile.
(a) What is the probability that there are no cracks that require repair in 5 miles of highway?
(b) What is the probability that at least one crack requires repair in mile of highway?
28
The probability that a person will die from a certain respiratory infection is 0.002. Find the probability that fewer than 5 of
the next 2000 so infected will die.
29
Suppose a life insurance company insures the lives of 5000 persons aged 42. If studies show the probability that any 42-years old
person will die in a given year to be 0.001, find the probability that the company will have to pay at least two claims during
a given year.
30
Suppose that, on average, 1 person in 1000 makes a numerical error in preparing his or her income tax return. If 10,000
forms are selected at random and examined, find the probability that 6, 7, or 8 of the forms contain an error.
HYPERGEOMETRIC & MULTIVARIATE HYPERGEOMETRIC DISTRIBUTION
31
A homeowner plants 6 bulbs selected at random from a box containing 5 tulip bulbs and 4 daffodil bulbs. What is the
probability that he planted 2 daffodil bulbs and 4 tulip bulbs?
32
To avoid detection at customs, a traveler places 6 narcotic tablets in a bottle containing 9 vitamin tablets that are similar in
appearance. If the customs official selects 3 of the tablets at random for analysis, what is the probability that the traveler
will be arrested for illegal possession of narcotics?
Shumaila Usman (Assistant professor)
(Mathematics Department)
NED UNIVERSITY OF ENGINEERING & TECHNOLOGY
SE (EE) SPRING SEMESTER 2024
CHAPTER 4 & 5
DISCRETE RANDOM VARIABLE/DISCRETE PROBABILITY DISTRIBUTION
Handout #2 P&S (MT-331)
33
From a lot of 10 missiles, 4 are selected at random and fired. If the lot contains 3 defective missiles that will not fire, what is
the probability that
(a) All 4 will fire?
(b) At most 2 will not fire?
34
What is the probability that a waitress will refuse to serve alcoholic beverages to only 2 minors if she randomly checks the
IDs of 5 among 9 students, 4 of whom are minors?
35
A company is interested in evaluating its current inspection procedure for shipments of 50 identical items. The procedure is
to take a sample of 5 and pass the shipment if no more than 2 are found to be defective. What proportion of shipments
with 20% defectives will be accepted?
36
A foreign student club lists as its members 2 Canadians, 3 Japanese, 5 Italians, and 2 Germans. If a committee of 4 is
selected at random, find the probability that
(a) All nationalities are represented.
(b) All nationalities except Italian are represented.
37
An urn contains 3 green balls, 2 blue balls, and 4 red balls. In a random sample of 5 balls, find the probability that both blue
balls and at least 1 red ball are selected.
Shumaila Usman (Assistant professor)
(Mathematics Department)
PROBABILITY & STATISTICS (MT-331)
Department :SE(EE) BATCH 2022
HANDOUT#3
1. Find the mean, median and mode of each set of data & write Comments on the shape of the Distribution:
A) 24,90,10,78,50,35,90,33,46
B) 5,9,1,6,4,8,3,5,6,9,7,4,8,5,5,1
C) 100,98,155,110,99,123,140,145
2- Choose the correct stem and leaf plot which represent the following data
16 10 11 30 35 33 35 44 47
a) 1
2
3
4
0 1 6
0
0 3 5 5
4 7
b) 1 0 1 6
3 0 3 5 5
4 4 7
c) 1 0 1 6
2
3 0 3 5 5
4 4 7
d) 1 0 1 6
3 0 3 5
4
4 7
Q3-(a) Draw frequency distribution table with the help of graph?
(b) Calculate Q1 Q2 & Q3 & write comments with the help of Quartiles?
(C) Verify result with the help of Graph?
(d) Calculate mode?
(e) Calculate inter quartile range.Also calculate 9th Decile?
ShumailaUsman
Assistant professor
Mathematics Department
I) Calculate inter quartile range?
J) Calculate variance & coeffient of variance?
k) Calculate skewness & kurtosis & write comments on the shape of the distribution?
Q5 (a) with the help of box plot calculate five number summary?
(b) Calculate inter quartile range and quartile deviation?
Q6 (a) draw frequency distribution table with the help of graph?
(b) Calculate skewness & kurtosis with the help of moment about mean?
(c) Write comments on the shape of distribution?
(d) Calculate coeffient of variation?
(e ) Calculate 99% range of the given data?
ShumailaUsman
Assistant professor
Mathematics Department
Q7 (a) calculate inter quartile range?
(b) Calculate median?
stem
10
9
8
7
leaf
0
4 5 5 8
5 5 5
4 7
Q8 (a) With the help of histogram draw frequency distribution table
(b) Calculate mean median & mode?
(c) Calculate variance & 68% range of the given data?
(d) Write comments on the shape of distribution?
Q9-Thirty AA batteries were tested to determine how long they would last. The results, to the nearest minute, were
recorded as follows:
423, 369, 387, 411, 393, 394, 371, 377, 389, 409, 392, 408, 431, 401, 363, 391, 405, 382, 400, 381, 399, 415, 428,
422, 396, 372, 410, 419, 386, 390
a) Construct frequency distribution table?
b) Calculate cumulative frequency distribution?
c) Calculate relative frequency distribution table?
d) Draw O-Give curve & frequency polygon?
Q10-A survey was taken on Maple Avenue. In each of 20 homes, people were asked how many cars were registered to
their households. The results were recorded as follows:
1, 2, 1, 0, 3, 4, 0, 1, 1, 1, 2, 2, 3, 2, 3, 2, 1, 4, 0, 0
ShumailaUsman
Assistant professor
Mathematics Department
a)
b)
c)
d)
Construct frequency distribution table?
Calculate cumulative frequency distribution?
Calculate relative frequency distribution table?
Draw O-Give curve & frequency polygon?
Calculate interquartile range & mode?
Q12-Construct a frequency distribution for the following data Use seven intervals, starting with 30-39.
79
60
74
59
55
98
61
67
83
71
71
46
63
66
69
42
75
62
71
77
78
65
87
57
78
91
82
73
94
48
87
65
62
81
63
66
65
49
45
51
69
56
84
93
63
60
68
51
73
54
50
88
76
93
48
70
39
76
95
57
63
94
82
54
89
64
77
94
72
69
51
56
67
88
81
70
81
54
66
87
(a)-Draw frequency distribution for inclusive & exclusive series?
(b)-Calculate mean, median, mode, 3rd quartile, skewness & kurtosis of the given data?
(c)-Construct a Cumulative frequency curve (O-GIVE), histogram and super impose frequency polygon on it.
(d)-Find Coefficient of variance?
(e) Verify the result of histogram with the help of mode?
(f) Calculate quartile deviation, mean absolute deviation & their coeffients?
ShumailaUsman
Assistant professor
Mathematics Department
13 The following table gives the number of weeks needed to find a job for 25 older workers that lost their jobs as a result of
corporation downsizing. Find Five number summary (min., Q1, median, Q3, max.) of the data and Interpret your answer
13
13
17
7
22
22
26
17
13
14
16
7
6
18
20
10
17
11
10
15
16
8
16
21
11
14 Find the mean deviation of the set (a) 3, 7, 9, 5 and (b) 2.4, 1.6, 3.8, 4.1, 3.4.
For the set 8, 10, 9, 12, 4, 8, 2, find the mean deviation from the mean and five number summary?
15 Goals scored by two team A & B in a football season were as follows
ShumailaUsman
Assistant professor
Mathematics Department
No of goals scored in
Match (xi)
Number of Matches
0
1
2
3
A
27
9
8
5
B
17
9
6
5
Find out mean and standard deviation & tell which team may be considered more consistent
16 The distribution of SAT scores for a group of high school students has a first quartile score equal to 825 and a third
quartile score equal to 1125. Calculate the quartile deviation for the distribution of SAT scores for this group of high
school students.
17 Construct a frequency distribution for the following data Use seven intervals, starting with 30-39.
79
60
74
59
55
98
61
67
83
71
71
46
63
66
69
42
75
62
71
77
78
65
87
57
78
91
82
73
94
48
87
65
62
81
63
66
65
49
45
51
69
56
84
93
63
60
68
51
73
54
50
88
76
93
48
70
39
76
95
57
63
94
82
54
89
64
77
94
72
69
51
56
67
88
81
70
81
54
66
87
(a)-Draw frequency distribution for inclusive & exclusive series?
rd
(b)-Calculate mean, median, mode,3 quartile , skewness & kurtosis of the given data?
(c)-Construct a c.f curve (ogive), histogram and super impose frequency polygon on it.
(d)-Find Coefficient of variance?
(e) Verify the result of histogram with the help of mode?
(f) Calculate quartile deviation, mean absolute deviation & their coeffients?
ShumailaUsman
Assistant professor
Mathematics Department
ShumailaUsman
Assistant professor
Mathematics Department
NED UNIVERSITY OF ENGINEERING & TECHNOLOGY
SE (EE) SPRING SEMESTER
Handout #4 P&S (MT-331)
CONTINUOUS RANDOM VARIABLE
1-
For a laboratory assignment, if the equipment is working, the density function of the observed outcome
X is
Find the variance and standard deviation of X.
2-
The total time, measured in units of 100 hours, that a teenager runs her hair dryer over a period of one
year is a continuous random variable X that has the density function
Evaluate the mean of the random variable Y =
expended annually.
3-
The total number of hours, measured in units of 100 hours, that a family runs a vacuum cleaner over a
Period of one year is a continuous random variable X that has the density function. Find the probability that over
a period of one year, a family runs their vacuum cleaner
I.
II.
4-
5-
6-
+ 39X, where Y is equal to the number of kilowatt hours
Less than 120 hours.
Between 50 and 100 hours.
A continuous random variable X that can assume values between x = 2 and x = 5 has a density function given by
f(x) = 2(1 + x)/27. Find
I.
P(X <4)
II.
P (3 ≤X <4).
III.
Find F(x), and use it to evaluate P (3 ≤X <4).
Consider the density function
I.
Evaluate k.
II.
Find F(x) and use it to evaluate P(0.3 < X < 0.6).
Let X be the random variable that denotes the life in hours of a certain electronic device. The probability density
function is
Find the expected life of this type of device.
Shumaila Usman
Assistant Professor (Mathematics Department)