Deck 6: Random Variables and Discrete Probability Distributions

Question 1

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 )+ 5, and V ( X + 5)= V ( X )+ 25.

Answer 1

False

Question 2

The length of time for which an apartment in a large complex remains vacant is a discrete random variable.

Answer 2

False

Question 3

The binomial random variable is the number of successes that occur in a fixed period of time.

Answer 3

False

Question 4

The number of homeless people in Boston is an example of a discrete random variable.

Question 5

For a random variable X , E ( X + 2)- 5 = E ( X )- 3, where E refers to the expected value.

Question 6

The amount of milk consumed by a baby in a day is an example of a discrete random variable.

Question 7

For a random variable X , if V ( cX )= 4 V ( X ), where V refers to the variance, then c must be 2.

Question 8

The binomial distribution deals with consecutive trials, each of which has two possible outcomes.

Question 9

A continuous variable may take on any value within its relevant range even though the measurement device may not be precise enough to record it.

Question 10

The time required to drive from New York to New Mexico is a discrete random variable.

Question 11

The binomial distribution deals with consecutive trials, each of which has two possible outcomes.

Question 12

A random variable is a function or rule that assigns a number to each outcome of an experiment.

Question 13

The number of home insurance policy holders is an example of a discrete random variable

Question 14

For a random variable X , V ( X + 3)= V ( X + 6), where V refers to the variance.

Question 15

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.

Question 16

The binomial probability distribution is a discrete probability distribution.

Question 17

The variance of a binomial distribution for which n = 50 and p = 0.20 is 8.0.

Question 18

The mean of a discrete probability distribution for X is the sum of all possible values of X , divided by the number of possible values of X .

Question 19

Faculty rank (professor, associate professor, assistant professor, and lecturer)is an example of a discrete random variable.

Question 20

Another name for the mean of a probability distribution is its expected value.

Question 21

If X is a binomial random variable with n = 25, and p = 0.25, then P ( X = 25)= 1.0.

Question 22

The mean of a Poisson distribution, where m is the average number of successes occurring in a specified interval, is m .

Question 23

The largest value that a Poisson random variable X can have is n .

Question 24

A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities, is called a(n):

A)discrete probability distribution.
B)discrete random variable.
C)expected value of a discrete random variable.
D)None of these choices.

Question 25

In a Poisson distribution, the mean and standard deviation are equal.

Question 26

The standard deviation of a binomial random variable X is given by the formula s ² = np (1 - p ), where n is the number of trials, and p is the probability of success.

Question 27

The Poisson random variable is a discrete random variable with infinitely many possible values.

Question 28

If the probability of success p remains constant in a binomial distribution, an increase in n will not change the mean.

Question 29

If the probability of success p remains constant in a binomial distribution, an increase in n will increase the variance.

Question 30

The Poisson probability distribution is a continuous probability distribution.

Question 31

In a Poisson distribution, the variance and standard deviation are equal.

Question 32

The number of customers making a purchase out of 30 randomly selected customers has a Poisson distribution.

Question 33

In a Poisson distribution, the mean and variance are equal.

Question 34

The expected number of heads in 250 tosses of an unbiased coin is 125.

Question 35

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.

Question 36

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)None of these choices.

Question 37

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 distribution.
C)a random variable.
D)None of these choices.

Question 38

The number of accidents that occur at a busy intersection in one month is an example of a Poisson random variable.

Question 39

The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.

Question 40

The number of female customers out of a random sample of 100 customers arriving at a department store has a binomial distribution.

Question 41

In a Poisson distribution, the:

A)mean equals the standard deviation.
B)median equals the standard deviation.
C)mean equals the variance.
D)None of these choices.

Question 42

The expected number of heads in 100 tosses of an unbiased coin is

A)25
B)50
C)75
D)100

Question 43

If n = 20 and p = 0.70, then the standard deviation of the binomial distribution is

A)0.14
B)2.05
C)14.0
D)14.7

Question 44

Which of the following is a continuous random variable?

A)The number of employees of an automobile company.
B)The amount of milk produced by a cow in one 24-hour period.
C)The number of gallons of milk sold at Albertson's grocery store last week.
D)None of these choices.

Question 45

Which of the following are required conditions for the distribution of a discrete random variable X that can assume values x_i ?

A)0 £ p ( x_i )£ 1     for all x_i
B)
C)Both a and b are required conditions.
D)Neither a nor b are required conditions.

Question 46

Which of the following about the binomial distribution is not a true statement?

A)The probability of success must be constant from trial to trial.
B)The random variable of interest is continuous.
C)Each outcome may be classified as either "success" or "failure".
D)Each outcome is independent of the other.

Question 47

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

Question 48

Which of the following cannot have a Poisson distribution?

A)The length of 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.

Question 49

The Sutton police department must write, on average, 6 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.5 tickets per day. Interpret the value of the mean.

A)The mean has no interpretation.
B)The expected number of tickets written would be 6.5 per day.
C)Half of the days have less than 6.5 tickets written and half of the days have more than 6.5 tickets written.
D)The number of tickets that is written most often is 6.5 tickets per day.

Question 50

In the notation below, X is the random variable, E and V refer to the expected value and variance, respectively. Which of the following is false?

A)E (3 X )= 3 E ( X )
B)V (2)= 0
C)E ( X + 1)= E ( X )+ 1
D)All of these choices are true.

Question 51

Which of the following is a discrete random variable?

A)The Dow Jones Industrial average.
B)The volume of water in Michigan Lakes.
C)The time it takes you to drive to school.
D)The number of employees of a soft drink company.

Question 52

Which of the following is not a characteristic of a binomial experiment?

A)Each trial results in two or more outcomes.
B)There is a sequence of identical trials.
C)The trials are independent of each other.
D)The probability of success p is the same from one trial to another.

Question 53

Which of the following is not a required condition for the distribution of a discrete random variable X that can assume values x_i ?

A)0 £ p ( x_i )£ 1     for all x_i
B)
C)p ( x_i )> 1     for all x_i
D)All of these choices are true.

Question 54

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.

Question 55

If n = 10 and p = 0.60, then the mean of the binomial distribution is

A)0.06
B)2.65
C)6.00
D)5.76

Question 56

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 true?

A)V ( c )= 0
B)V ( X + c )= V ( X )+ c
C)V ( cX )= c ² V ( X )
D)None of these choices.

Question 57

The variance of a binomial distribution for which n = 100 and p = 0.20 is:

A)100
B)80
C)20
D)16

Question 58

A lab at the DeBakey Institute orders 150 rats a week for each of the 52 weeks in the year for experiments that the lab conducts. Suppose the mean cost of rats used in lab experiments turned out to be $20.00 per week. Interpret this value.

A)Most of the weeks resulted in rat costs of $20.00
B)The median cost for the distribution of rat costs is $20.00
C)The expected or average costs for all weekly rat purchases is $20.00
D)The rat cost that occurs more often than any other is $20.00

Question 59

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)expected value of a discrete random variable.
D)expected value of a continuous random variable.

Question 60

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

Question 61

On the average, 1.6 customers per minute arrive at any one of the checkout counters of Sunshine food market. What type of probability distribution can be used to find out the probability that there will be no customers arriving at a checkout counter in 10 minutes?

A)Poisson distribution
B)Normal distribution
C)Binomial distribution
D)None of these choices.

Question 62

The dean of students conducted a survey on campus. Grade point average (GPA)is an example of a(n)____________________ random variable.

Question 63

The probability P ( X £ x )is called a(n)____________________ probability. The binomial table reports these probabilities.

Question 64

An auto insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for automobile insurance. The number of claims a person has made in the last 3 years is an example of a(n)____________________ random variable.

Question 65

An auto insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for automobile insurance. A person's age is an example of a(n)____________________ random variable.

Question 66

The trials in a binomial experiment are ____________________, meaning the outcome of one trial does not affect the outcomes of any other trials.

Question 67

The number of days that a microcomputer goes without a breakdown is an example of a(n)____________________ random variable.

Question 68

A(n)____________________ random variable is one whose values are countable.

Question 69

A community college has 150 word processors. The probability that any one of them will require repair on a given day is 0.025. To find the probability that exactly 25 of the word processors will require repair, one will use what type of probability distribution?

A)Normal distribution
B)Poisson distribution
C)Binomial distribution
D)None of these choices.

Question 70

The probability of a success in a binomial experiment is denoted by ____________________.

Question 71

A motorcycle insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for motorcycle insurance. How long a person has been a licensed rider is an example of a(n)____________________ random variable.

Question 72

The amount of time that a microcomputer is used per week is an example of a(n)____________________ random variable.

Question 73

A motorcycle insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for motorcycle insurance. The number of tickets a person has received in the last 3 years is an example of a(n)____________________ random variable.

Question 74

The probability of a failure in a binomial experiment is denoted by ____________________.

Question 75

In each trial of a binomial experiment, there are ____________________ possible outcomes.

Question 76

A motorcycle insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for motorcycle insurance. The distance a person rides in a year is an example of a(n)____________________ random variable.

Question 77

A binomial experiment consists of a(n)____________________ number of trials, n .

Question 78

A(n)____________________ random variable is one whose values are uncountable.

Question 79

The variance of a binomial distribution is equal to ____________________.

Question 80

The mean of a binomial distribution is equal to ____________________.