Deck 7: Random Variables and Discrete Probability Distributions

Question 1

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.

Answer 1

True

Question 2

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

Answer 2

True

Question 3

The Poisson probability distribution is a continuous probability distribution.

Answer 3

False

Question 4

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.

Question 5

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

Question 6

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

Question 7

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

Question 8

A table,formula,or graph that shows all possible values a random variable can assume,together with their associated probabilities,is referred to as probability distribution.

Question 9

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

Question 10

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 11

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

Question 12

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

Question 13

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

Question 14

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

Question 15

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

Question 16

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

Question 17

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

Question 18

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

Question 19

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

Question 20

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

Question 21

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

Question 22

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 23

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 24

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 25

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 26

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

Question 27

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

Question 28

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 29

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 30

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 31

A function or rule that assigns a numerical value to each outcome of an experiment is called:

A)a sample space.
B)a probability distribution.
C)a random variable.
D)None of these choices.

Question 32

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

Question 33

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 34

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)= c² V(X)
D)None of these choices.

Question 35

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 36

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

Question 37

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

Question 38

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 39

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

Question 40

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 41

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 42

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 43

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

Question 44

In a(n)____________________ experiment,the probability of a success in an interval is the same for all equal-sized intervals.

Question 45

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 46

In a Poisson experiment,the probability of a success in an interval is ____________________ to the size of the interval.

Question 47

The ____________________ of a Poisson distribution is the rate at which successes occur for a given period of time or interval of space.

Question 48

A Poisson random variable is the number of successes that occur in a period of ____________________ or an interval of ____________________ in a Poisson experiment.

Question 49

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

Question 50

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 51

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

Question 52

In Poisson experiment,the probability of more than one success in an interval approaches ____________________ as the interval becomes smaller.

Question 53

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 54

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 55

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

Question 56

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 57

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 58

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

Question 59

In a Poisson experiment,the number of successes that occur in any interval of time is ____________________ of the number of success that occur in any other interval.

Question 60

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 61

Determine which of the following are not valid probability distributions,and explain why not.

Question 62

The possible values of a Poisson random variable start at ____________________.

Question 63

Number of Motorcycles
The probability distribution of a discrete random variable X is shown below,where X represents the number of motorcycles owned by a family. ​ ​
{Number of Motorcycles Narrative} Find the standard deviation of X.

Question 64

Number of Horses ​
The random variable X represents the number of horses per family in a rural area in Iowa,with the probability distribution: p(x)= 0.05x,x = 2,3,4,5,or 6. ​ ​
{Number of Horses Narrative} Find the expected number of horses per family.

Question 65

Blackjack
The probability distribution of a random variable X is shown below,where X represents the amount of money (in $1,000s)gained or lost in a particular game of Blackjack. ​ ​
{Blackjack Narrative} Find the following probabilities:
a.P(X ≤ 0)
b.P(X > 3)
c.P(0 ≤ X ≤ 4)
d.P(X = 5)

Question 66

Gym Visits
Let X represent the number of times a student visits a gym in a one month period.Assume that the probability distribution of X is as follows: ​ ​
{Gym Visits Narrative} Find the mean μ and the standard deviation σ of this distribution.

Question 67

Gym Visits
Let X represent the number of times a student visits a gym in a one month period.Assume that the probability distribution of X is as follows: ​ ​
{Gym Visits Narrative} What is the probability that the student visits the gym at least once in a month?

Question 68

Number of Horses ​
The random variable X represents the number of horses per family in a rural area in Iowa,with the probability distribution: p(x)= 0.05x,x = 2,3,4,5,or 6. ​ ​
{Number of Horses Narrative} Find the variance and standard deviation of X.

Question 69

A Poisson random variable is a(n)____________________ random variable.

Question 70

Number of Motorcycles
The probability distribution of a discrete random variable X is shown below,where X represents the number of motorcycles owned by a family. ​ ​
{Number of Motorcycles Narrative} Find the expected value of X.

Question 71

In the Poisson distribution,the ____________________ is equal to the variance.

Question 72

In the Poisson distribution,the mean is equal to the ____________________.

Question 73

Number of Motorcycles
The probability distribution of a discrete random variable X is shown below,where X represents the number of motorcycles owned by a family. ​ ​
{Number of Motorcycles Narrative} 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)

Question 74

Number of Motorcycles
The probability distribution of a discrete random variable X is shown below,where X represents the number of motorcycles owned by a family. ​ ​
{Number of Motorcycles Narrative} Apply the laws of expected value to find the following:
a.E(X²)
b.E(2X² + 5)
c.E(X − 2)²

Question 75

Number of Horses ​
The random variable X represents the number of horses per family in a rural area in Iowa,with the probability distribution: p(x)= 0.05x,x = 2,3,4,5,or 6. ​ ​
{Number of Horses Narrative} 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)

Question 76

Number of Horses ​
The random variable X represents the number of horses per family in a rural area in Iowa,with the probability distribution: p(x)= 0.05x,x = 2,3,4,5,or 6. ​ ​
{Number of Horses Narrative} Express the probability distribution in tabular form.

Question 77

Gym Visits
Let X represent the number of times a student visits a gym in a one month period.Assume that the probability distribution of X is as follows: ​ ​
{Gym Visits Narrative} Find the mean and the standard deviation of Y = 2X − 1.

Question 78

Number of Motorcycles
The probability distribution of a discrete random variable X is shown below,where X represents the number of motorcycles owned by a family. ​ ​
{Number of Motorcycles Narrative} Apply the laws of expected value and variance to find the following:
a.V(3X)
b.V(3X − 2)
c.V(3)
d.V(3X)− 2

Question 79

Blackjack The probability distribution of a random variable X is shown below,where X represents the amount of money (in $1,000s)gained or lost in a particular game of Blackjack. ​ ​
{Blackjack Narrative} Find the following values and indicate their units.
a.E(X)
b.V(X)
c.Standard deviation of X

Question 80

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 Albuquerque 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 175 families.
d.X = the number of households out of 10 surveyed that own a convection oven.
e.X = the time in minutes required to obtain service in a restaurant.