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# Ch07. 2320

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York University

Administrative Studies

ADMS 2300

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CHAPTER 7
RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS
MULTIPLE CHOICE QUESTIONS
In the following multiple-choice questions, please circle the correct answer.
1. The weighted average of the possible values that a random variable X can
assume, where the weights are the probabilities of occurrence of those
values, is referred to as the:
a. variance
b. standard deviation
c. expected value
d. covariance
ANSWER: c
2. The number of accidents that occur annually on a busy stretch of highway is
an example of:
a. a discrete random variable
b. a continuous random variable
c. a discrete probability distribution
d. a continuous probability distribution
ANSWER: a
3. A function or rule that assigns a numerical value to each simple event of an
experiment is called:
a. a sample space
b. a probability tree
c. a probability distribution
d. a random variable
ANSWER: d
4. If X and Y are any random variables, which of the following identities is not
always true?
a. E (X+Y) = E(X) + E(Y)
b. V(X+Y) = V(X) + V(Y)
c. E(4X+5Y) = 4E(X) + 5 E(Y)
d. V(4X+5Y) = 16V(X) + 25V(Y) + 40COV(X,Y)
ANSWER: b
5. If X and Y are random variables, the sum of all the conditional probabilities of
X given a specific value of Y will always be:
a. 0.0
94 95 Chapter Seven
b. 1.0
c. the average of the possible values of X
d. a value larger than zero but smaller than 1.0
ANSWER: b
6. A statistical measure of the strength of the linear relationship between two
random variables X and Y is referred to as the:
a. expected value
b. variance
c. covariance
d. standard deviation
ANSWER: c
7. A table, formula, or graph that shows all possible values a random variable
can assume, together with their associated probabilities is called a:
a. discrete probability distribution
b. continuous probability distribution
c. bivariate probability distribution
d. probability tree
ANSWER: a
8. If X and Y are random variables with E(X) = 5 and E(Y) = 8, then E(2X+3Y) is:
a. 34
b. 13
c. 18
d. 40
ANSWER: a
9. If X and Y are random variables with V(X) = 7.5, V(Y) = 6 and COV(X,Y) = 4,
then V(2X+3Y) is:
a. 33
b. 37
c. 88
d. 132
ANSWER: d
10. If X and Y are independent random variables, which of the following identities
is always true?
a. E(2X+3Y) = E(X) + E(Y) + 5
b. V(2X+3Y) =2V(X) + 3V(Y)
c. V(2X+3Y) = 4V(X) + 9V(Y)
d. E(2X+3Y) = 5E(X+Y)
ANSWER: c
11. If X and Y are any random variables with E(X) = 50, E(Y) = 6, E(XY) = 21,
V(X) = 9 and V(Y) = 10, then the relationship between X and Y is a :
a. strong positive relationship
b. strong negative relationship
c. weak positive relationship
d. weak negative relationship Random Variables and Discrete Probability96istributions
ANSWER: b
12. Which of the following is not a characteristic of a binomial experiment?
a. There is a sequence of identical trials
b. Each trial results in two or more outcomes.
c. The trials are independent of each other.
d. Probability of success p is the same from one trial to another.
ANSWER: b
13. The expected value, E(X), of a binomial probability distribution with n trials
and probability p of success is:
a. n + p
b. np(1-p)
c. np
d. n + p - 1
ANSWER: c
14. The Poisson random variable is a:
a. discrete random variable with infinitely many possible values
b. discrete random variable with finite number of possible values
c. continuous random variable with infinitely many possible values
d. continuous random variable with finite number of possible values
ANSWER: a
15. Given a Poisson random variable X, where the average number of successes
occurring in a specified interval is 1.8, then P(X = 0) is
a. 1.8
b. 1.3416
c. 0.1653
d. 6.05
ANSWER: c
16. Which probability distribution is appropriate when the events of interest
occur randomly, independently of one another, and rarely?
a. Binomial distribution
b. Poisson distribution
c. Any discrete probability distribution
d. Any continuous probability distribution
ANSWER: b
17. The expected number of heads in 100 tosses of an unbiased coin is
a. 30
b. 40
c. 50
d. 60
ANSWER: c
18. Which of the following cannot generate a Poisson distribution? 97 Chapter Seven
a. The number of children watching a movie
b. The number of telephone calls received by a switchboard in a specified
time period
c. The number of customers arriving at a gas station in Christmas day
d. The number of bacteria found in a cubic yard of soil
ANSWER: a
19. The Poisson probability distribution is used with
a. a discrete random variable
b. a continuous random variable
c. either a discrete or a continuous random variable, depending on the mean
d. either a discrete or a continuous random variable, depending on the
sample size
ANSWER: a
20. The variance of a binomial distribution for which n = 100 and p = 0.20 is:
a. 100
b. 80
c. 20
d. 16
ANSWER: d
21. Which of the following is (are) required condition(s) for the distribution of a
discrete random variable X that can assume values x ?
i
a. 0 ≤ p(xi) ≤1 for all i
b. ∑ p(xi) =1
allix
c. Both a and b are required conditions
d. Only b is a required condition
ANSWER: c
22. Which of the following is not a required condition for the distribution of a
discrete random variable X that can assume values xi?
a. 0 ≤ p(xi) ≤1 for all i
b. ∑ p(xi) =1
allix
c. p(x ) >1 for all x
i i
d. All of the above are not required conditions
ANSWER: c
23. The binomial probability distribution is used with
a. a discrete random variable
b. a continuous random variable
c. either a discrete or a continuous random variable, depending on the
variance Random Variables and Discrete Probabili98 Distributions
d. either a discrete or a continuous random variable, depending on the
sample size
ANSWER: a
24. Twenty five percent of the students in an English class 0f 100 are
international students. The standard deviation of this binomial distribution is
a. 25
b. 5
c. 18.75
d. 4.33
ANSWER: d
25. In the notation below, X is the random variable, c is a constant, and V refers
to the variance. Which of the following laws of variance is not correct?
a. V(c) = 0
b. V(X + c) = V(X)
c. V(X + c) = V(X) + c
d. V(cX) = c V(X)
ANSWER: c
TRUE/FALSE QUESTIONS
26. A random variable is a function or rule that assigns a number to each
outcome of an experiment.
ANSWER: T
27. The time required to drive from Detroit to Chicago is a discrete random
variable.
ANSWER: F
28. The binomial distribution deals with consecutive trials, each of which has two
possible outcomes.
ANSWER: T
29. The number of home insurance policy holders is an example of a discrete
random variable
ANSWER: T
30. The mean of a discrete probability distribution is given by the equation
μ =∑ x⋅ p(.)
ANSWER: T
31. Poisson distribution is appropriate to determine the probability of a given
number of defective items in a shipment.
ANSWER: F
32. The length of time for which an apartment in a large complex remains vacant
is a discrete random variable.
ANSWER: F 99 Chapter Seven
33. The number of homeless people in New York City is an example of a discrete
random variable.
ANSWER: T
34. Given that X is a discrete random variable, then the laws of expected value
and variance can be applied to show that E(X + 5) = E(X), and V(X+5) = V(X)
+ 25.
ANSWER: F
35. In a southern state, stabbings of inmates by other inmates at a state prison
have averaged 2.5 per month over the past year. Let X be the number of
inmate stabbings per month, and assume that a Poisson distribution is
appropriate for this situation. Then, P(X = 3) = 0.833.
ANSWER: F
36. The Poisson distribution is applied to events for which the probability of
occurrence over a given span of time, space, or distance is very small.
ANSWER: T
37. For a given probability of success p, the binomial distribution tends to take
on more of a bell shape as the number of trials n increases.
ANSWER: T
38. A table, formula, or graph that shows all possible values a random variable
can assume, together with their associated probabilities, is referred to as
discrete probability distribution.
ANSWER: T
39. Any discrete distribution is appropriate when the events of interest occur
randomly, independently of one another, and rarely.
ANSWER: F
40. The Poisson random variable is a discrete random variable with infinitely
many possible values.
ANSWER: T
41. The mean of a Poisson distribution, where μ is the average number of
successes occurring in a specified interva., is
ANSWER: T
42. The number of accidents that occur at a busy intersection is an example of a
Poisson random variable.
ANSWER: T
43. The variance of a binomial distribution for which n = 50 and p = 0.20 is 8.0.
ANSWER: T
44. The expected number of heads in 250 tosses of an unbiased coin is 125.
ANSWER: T Random Variables and Discrete Probability100stributions
45. The binomial random variable is the number of successes that occur in a
period of time or an interval of space.
ANSWER: F
46. If X is a binomial random variable with n = 25, and p = 0.25, then P(X = 25)
= 1.0.
ANSWER: F
47. The number of students that use a computer lab during one day is an
example of either a continuous or a discrete random variable, depending on
the number of the students.
ANSWER: F
48. The Poisson probability distribution is a continuous probability distribution.
ANSWER: F
49. The binomial probability distribution is a discrete probability distribution.
ANSWER: T
50. The standard deviation of a binomial random variable X is given by the
formula σ = np(1− p) where n is the number of trials, and p is the probability
of success.
ANSWER: F
51. The temperature of the room in which you are taking this test is a continuous
quantitative variable.
ANSWER: T
TEST QUESTIONS
52. For each of the following random variables, indicate whether the variable is
discrete or continuous, and specify the possible values that it can assume.
a. X = the number of traffic accidents in Phoenix on a given day.
b. X = the amount of weight lost in a month by a randomly selected dieter.
c. X = the average number of children per family in a random sample of 100
families.
d. X = the number of households out of 10 surveyed that own a microwave
oven.
e. X = the time in minutes required to obtain service in a restaurant.
ANSWERS:
a. discrete; x = 0, 1, 2, 3, . . .
b. continuous; –∞ < x < ∞
c. continuous; x≥ 0
d. discrete; x = 0, 1, 2, . . . , 10
e. continuous; x > 0 101 Chapter Seven
QUESTIONS 53 THROUGH 56 ARE BASED ON THE FOLLOWING INFORMATION:
The probability distribution of a discrete random variable X is shown below.
x 0 1 2 3
p(x) 0.25 0.40 0.20 0.15
53. Find the following probabilities:
a. P(X > 1)
b. P(X 2)
≤
c. P(1 ≤ X≤ 2)
d. P(0 < X < 1)
e. P(1 ≤ X<3)
ANSWERS:
a. 0.35
b. 0.85
c. 0.60
d. 0.00
e. 0.60
54. Find the expected value, the variance, and the standard deviation of X.
ANSWERS:
σ
E(X) = 1.25, V(X) = 0.9875, and x= 0.9937
2
55. a. Find E( X )
b. Find E(2 X 2+ 5)
c. Find E ( X − 2)
ANSWERS:
a. 2.55
b. 10.1
c. 1.55
56. a. Find E(3X-2)
b. Find V(3X-2)
c. Find E(2 2+ 3X - 1)
X
ANSWERS:
a. 1.75
b. 8.8875
c. 7.85
QUESTIONS 57 AND 58 ARE BASED ON THE FOLLOWING INFORMATION: Random Variables and Discrete Probability102stributions
Consider a random variable X with the following probability distribution:
p(x) = 0.05x, x = 2, 3, 4, 5, or 6
57. Express the probability distribution in tabular form.
ANSWER:
x 2 3 4 5 6
p(x) 0.10 0.15 0.20 0.25 0.30
58. Find the following probabilities:
a. P(X 4)
≥
b. P(X > 4)
c. P(3≤ X ≤ 5)
d. P(2 < X < 4)
e. P(X = 4.5)
ANSWERS
a. 0.75
b. 0.55
c. 0.60
d. 0.15
e. 0.00
59. Determine which of the following are not valid probability distributions, and
explain why not.
a.
x 0 1 2 3
p(x) 0.15 0.25 0.35 0.45
b.
x 2 3 4 5
p(x) - 0.40 0.50 0.25
0.10
c.
x -2 -1 0 1 2
p(x) 0.10 0.20 0.40 0.20 0.10
ANSWERS:
Table (a) is not a valid probability distribution because the probabilities don’t
sum to one, and Table (b) is not valid because it contains a negative
probability. Table (c) is a valid probability distribution. 103 Chapter Seven
QUESTIONS 60 THROUGH 63 ARE BASED ON THE FOLLOWING INFORMATION:
The probability distribution of a random variable X is shown below.
x -4 0 4 8
p(x) 0.15 0.25 0.20 0.40
60. Find the following probabilities:
a. P(X ≤ 0)
b. P(X > 3)
c. P(0 ≤ X≤ 4)
d. P(X = 5)
ANSWERS:
a. 0.40
b. 0.60
c. 0.45
d. 0.00
61. a. Find E(X).
b. Find V(X).
ANSWERS:
a. 3.40
b. 19.64
62. Find the expected value of Y =X 2- 2X +1
ANSWER:
25.40
63. a. Find E(3X-4).
b. Find V(3X-4).
ANSWERS:
a. 6.20
b. 176.76
QUESTIONS 64 THROUGH 66 ARE BASED ON THE FOLLOWING INFORMATION:
Let X represent the number of children in an Egyptian household. The probability
distribution
of X is as follows:
x 1 2 3 4 5
p(x) 0.25 0.33 0.17 0.15 0.10
64. What is the probability that a randomly selected Egyptian household will
have Random Variables and Discrete Probability 104tributions
a. more than 3 children?
b. between 3 and 5 children?
c. fewer than 4 children?
ANSWERS:
a. 0.25
b. 0.42
c. 0.75
65. Find the expected number of children in a randomly selected Egyptian
household.
ANSWER:
2.52
66. Find the standard deviation of the number of children in an Egyptian
household.
ANSWER:
1.2844
QUESTIONS 67 THROUGH 70 ARE BASED ON THE FOLLOWING INFORMATION:
Let X represent the number of times a student visits a bookstore in a 1-month
period. Assume
that the probability distribution of X is as follows:
x 0 1 2 3
p(x) 0.05 0.25 0.50 0.20
67. Find the mean μ and the standard deviationσ of this distribution.
ANSWER:
μx =1.85, and σ x= 0.792
68. Find the mean and the standard deviation of Y = 2X – 1.
ANSWER:
μy = 2.70, and σ y= 1.584 105 Chapter Seven
69. What is the probability that the student visits the bookstore at least once in a

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