Deck 6: Continuous Probability Distributions

Question 1

The cumulative distribution function F(x)of a continuous random variable X with the probability density function f(x)is which of the following?

A)The area under f over all values x
B)The area under f over all values that are x or less
C)The area under f over all values that are x or more
D)The area under f over all non-negative values that are x or less

Answer 1

The area under f over all values that are x or less

Question 2

The continuous uniform distribution describes a random variable,defined on the interval [a,b],that has an equally likely chance of assuming values within any subinterval of [a,b] with the same length.

Answer 2

True

Question 3

The probability density function of a continuous uniform distribution is positive for all values between -∞ and +∞.

Answer 3

False

Question 4

A continuous random variable is characterized by uncountable values and can take on any value within an interval.

Question 5

Which of the following is correct?

A)A continuous random variable has a probability density function but not a cumulative distribution function.
B)A discrete random variable has a probability mass function but not a cumulative distribution function.
C)A continuous random variable has a probability mass function,and a discrete random variable has a probability density function.
D)A continuous random variable has a probability density function,and a discrete random variable has a probability mass function.

Question 6

The standard normal distribution is a normal distribution with a mean equal to zero and a standard deviation equal to one.

Question 7

The mean of a continuous uniform distribution is simply the average of the upper and lower limits of the interval on which the distribution is defined.

Question 8

Which of the following does not represent a continuous random variable?

A)Height of oak trees in a park.
B)Heights and weights of newborn babies.
C)Time of a flight between Chicago and New York.
D)The number of customer arrivals to a bank between 10 am and 11 am.

Question 9

Just as in the case of the continuous uniform distribution,the probability density function of the normal distribution may be easily used to compute probabilities.

Question 10

The letter Z is used to denote a random variable with any normal distribution.

Question 11

The mean and standard deviation of the continuous uniform distribution are equal.

Question 12

The probability density function of a continuous random variable can be regarded as a counterpart of the probability mass function of a discrete random variable.

Question 13

The probability density function of a normal distribution is in general characterized by being symmetric and bell-shaped.

Question 14

The standard normal table is also referred to as the z table.

Question 15

Examples of random variables that closely follow a normal distribution include the age and the class year designation of a college student.

Question 16

We are often interested in finding the probability that a continuous random variable assumes a particular value.

Question 17

Which of the following is not a characteristic of a probability density function f(x)?

A)f(x)≥ 0 for all values of x.
B)f(x)is symmetric around the mean.
C)The area under f(x)over all values of x equals one.
D)f(x)becomes zero or approaches zero if x increases to +infinity or decreases to -infinity.

Question 18

Cumulative distribution functions can only be used to compute probabilities for continuous random variables.

Question 19

The cumulative distribution function is denoted and defined as which of the following?

A)f(x)and f(x)= P(X ≤ x)
B)f(x)and f(x)= P(X ≥ x)
C)F(x)and F(x)= P(X ≤ x)
D)F(x)and F(x)= P(X ≥ x)

Question 20

Given that the probability distribution is normal,it is completely described by its mean μ > 0 and its standard deviation σ > 0.

Question 21

Let X be normally distributed with mean μ and standard deviation σ > 0.Which of the following is true about the z value corresponding to a given x value?

A)A positive z = (x - μ)/σ indicates how many standard deviations x is above μ.
B)A negative z = (x - μ)/σ indicates how many standard deviations x is below μ.
C)The z value corresponding to x = μ is zero.
D)All of the above.

Question 22

The probability P(Z > 1.28)is closest to ____.

A)-0.10
B)0.10
C)0.20
D)0.90

Question 23

Suppose the average price of gasoline for a city in the United States follows a continuous uniform distribution with a lower bound of $3.50 per gallon and an upper bound of $3.80 per gallon.What is the probability a randomly chosen gas station charges more than $3.70 per gallon?

A)0.3000
B)0.3333
C)0.6667
D)1.0000

Question 24

It is known that the length of a certain product X is normally distributed with μ = 20 inches and σ = 4 inches.How is the probability related to ?

A) is greater than .
B) is smaller than .
C) is the same as .
D)No comparison can be made with the given information.

Question 25

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds.Find the mean and standard deviation of the waiting time.

A)115 seconds and 49.07 seconds
B)1.15 minutes and 0.4907 minutes
C)1.15 minutes and 24.08333 (minute)²
D)115 seconds and 2408.3333 (second)²

Question 26

The probability that a normal random variable is less than its mean is ___.

A)0.0
B)0.5
C)1.0
D)Cannot be determined

Question 27

It is known that the length of a certain product X is normally distributed with μ = 20 inches.How is the probability related to ?

A) is greater than .
B) is smaller than .
C) is the same as .
D)No comparison can be made with the given information.

Question 28

A continuous random variable has the uniform distribution on the interval [a,b] if its probability density function f(x)__________.

A)Is symmetric around its mean
B)Is bell-shaped between a and b
C)Is constant for all x between a and b,and 0 otherwise
D)Asymptotically approaches the x axis when x increases to +∞ or decreases to -∞

Question 29

The probability P(Z < -1.28)is closest to ____.

A)-0.10
B)0.10
C)0.20
D)0.90

Question 30

The time of a call to a technical support line is uniformly distributed between 2 and 10 minutes.What are the mean and variance of this distribution?

A)6 minutes and 2.3094 (minutes)²
B)6 minutes and 5.3333 (minutes)²
C)6 minutes and 5.3333 minutes
D)8 minutes and 2.3094 minutes

Question 31

It is known that the length of a certain product X is normally distributed with μ = 20 inches.How is the probability related to ?

A) is greater than .
B) is smaller than .
C) is the same as .
D)No comparison can be made with the given information.

Question 32

The height of the probability density function f(x)of the uniform distribution defined on the interval [a,b] is ____________.

A)1/(b - a)between a and b,and zero otherwise
B)(b - a)/2 between a and b,and zero otherwise
C)(a + b)/2 between a and b,and zero otherwise
D)1/(a + b)between a and b,and zero otherwise

Question 33

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds.What is the probability a rider must wait between 1 minute and 1.5 minutes?

A)0.1765
B)0.3529
C)0.5294
D)0.8824

Question 34

What does it mean when we say that the tails of the normal curve are asymptotic to the x axis?

A)The tails get closer and closer to the x axis but never touch it.
B)The tails gets closer and closer to the x axis and eventually touch it.
C)The tails get closer and closer to the x axis and eventually cross this axis.
D)The tails get closer and closer to the x axis and eventually become this axis.

Question 35

An analyst is forecasting net income for Excellence Corporation for the next fiscal year.Her low-end estimate of net income is $250,000,and her high-end estimate is $350,000.Prior research allows her to assume that net income follows a continuous uniform distribution.The probability that net income will be greater than or equal to $337,500 is _______.

A)12.5%
B)29.6%
C)87.5%
D)96.4%

Question 36

It is known that the length of a certain product X is normally distributed with μ = 20 inches.How is the probability related to ?

A) is greater than .
B) is smaller than .
C) is the same as .
D)No comparison can be made with the given information.

Question 37

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds.What is the probability a rider waits less than two minutes?

A)0.4706
B)0.5294
C)0.6000
D)0.7059

Question 38

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds.What is the probability a rider must wait more than 1.5 minutes?

A)0.3529
B)0.4500
C)0.5294
D)0.6471

Question 39

How many parameters are needed to fully describe any normal distribution?

A)1
B)2
C)3
D)4

Question 40

Alex is in a hurry to get to work and is rushing to catch the bus.She knows that the bus arrives every six minutes during rush hour,but does not know the exact times the bus is due.She realizes that from the time she arrives at the stop,the amount of time that she will have to wait follows a uniform distribution with a lower bound of 0 minutes and an upper bound of six minutes.What is the probability that she will have to wait more than two minutes?

A)0.1667
B)0.3333
C)0.6667
D)1.0000

Question 41

A hedge fund returns on average 26% per year with a standard deviation of 12%.Using the empirical rule,approximate the probability the fund returns over 50% next year.

A)0.5%
B)1%
C)2.5%
D)5%

Question 42

Let X be normally distributed with mean µ = 250 and standard deviation σ = 80.Find the value x such that P(X ≤ x)= 0.9394.

A)-1.55
B)1.55
C)126
D)374

Question 43

Let X be normally distributed with mean µ = 250 and standard deviation σ = 80.Find the value x -such that P(X ≤ x)= 0.0606.

A)-1.55
B)1.55
C)126
D)374

Question 44

Find the z value such that .

A)z = -1.645
B)z = -1.96
C)z = 1.645
D)z = 1.96

Question 45

Find the z value such that .

A)z = -1.33
B)z = 0.1814
C)z = 0.8186
D)z = 1.33

Question 46

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour.Find the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.

A)0.3085
B)0.3830
C)0.6170
D)0.6915

Question 47

Find the probability P(-1.96 ≤ Z ≤ 1.96).

A)0.0500
B)0.9500
C)0.9750
D)1.9500

Question 48

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour.Find the probability that it would take exactly 3.7 hours to construct a soapbox derby car.

A)0.0000
B)0.5000
C)0.7580
D)0.2420

Question 49

The starting salary of an administrative assistant is normally distributed with a mean of $50,000 and a standard deviation of $2,500.We know that the probability of a randomly selected administrative assistant making a salary between μ - x and μ + x is 0.7416.Find the salary range referred to in this statement.

A)$42,825 to $52,825
B)$42,825 to $57,175
C)$47,175 to $52,825
D)$47,175 to $57,175

Question 50

You work in marketing for a company that produces work boots.Quality control has sent you a memo detailing the length of time before the boots wear out under heavy use.They find that the boots wear out in an average of 208 days,but the exact amount of time varies,following a normal distribution with a standard deviation of 14 days.For an upcoming ad campaign,you need to know the percent of the pairs that last longer than six months-that is,180 days.Use the empirical rule to approximate this percent.

A)2.5%
B)5%
C)95%
D)97.5%

Question 51

Sarah's portfolio has an expected annual return at 8%,with an annual standard deviation at 12%.If her investment returns are normally distributed,then in any given year Sarah has approximately ______________________.

A)A 50% chance that the actual return will be greater than 8%
B)About a 68% chance that the actual return will fall within 4% and 20%
C)About a 68% chance that the actual return will fall within -20% and 20%
D)About a 95% chance that the actual return will fall within -4% and 28%.

Question 52

Find the probability P(-1.96 ≤ Z ≤ 0).

A)0.0250
B)0.0500
C)0.4750
D)0.5250

Question 53

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour.Find the probability that it would take more than five hours to construct a soapbox derby car.

A)0
B)0.0228
C)0.4772
D)0.9772

Question 54

The stock price of a particular asset has a mean and standard deviation of $58.50 and $8.25,respectively.Use the normal distribution to compute the 95^th percentile of this stock price.

A)-1.645
B)1.645
C)44.93
D)72.07

Question 55

The salary of teachers in a particular school district is normally distributed with a mean of $50,000 and a standard deviation of $2,500.Due to budget limitations,it has been decided that the teachers who are in the top 2.5% of the salaries would not get a raise.What is the salary level that divides the teachers into one group that gets a raise and one that doesn't?

A)-1.96
B)1.96
C)45,100
D)54,900

Question 56

For any normally distributed random variable with mean μ and standard deviation σ,the percent of the observations that fall between and is closest to ______.

A)68%
B)68.26%
C)95%
D)95.44%

Question 57

For any normally distributed random variable with mean μ and standard deviation σ,the proportion of the observations that fall outside the interval [μ - σ,μ + σ] is closest to _____.

A)0.0466
B)0.3174
C)0.8413
D)0.1687

Question 58

Let X be normally distributed with mean µ = 25 and standard deviation σ = 5.Find the value x such that P(X ≥ x)= 0.1736.

A)-0.94
B)0.94
C)20.30
D)29.70

Question 59

If X has a normal distribution with and ,then the probability can be expressed in terms of a standard normal variable Z as _______.

A)
B)
C)
D)

Question 60

An investment consultant tells her client that the probability of making a positive return with her suggested portfolio is 0.90.What is the risk,measured by standard deviation,that this investment manager has assumed in his calculation if it is known that returns from her suggested portfolio are normally distributed with a mean of 6%?

A)1.28%
B)4.69%
C)6.00%
D)10.0%

Question 61

Exhibit 6-2.Gold miners in Alaska have found,on average,12 ounces of gold per 1000 tons of dirt excavated with a standard deviation of 3 ounces.Assume the amount of gold found per 1000 tons of dirt is normally distributed. Refer to Exhibit 6-2.What is the probability the miners find more than 16 ounces of gold in the next 1000 tons of dirt excavated?

A)0.0918
B)0.4082
C)0.5918
D)0.9082

Question 62

Exhibit 6-1.You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough.The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F. Refer to Exhibit 6-1.Above what temperature must the sleeping bag be suited such that the temperature will be too cold only 5% of the time?

A)-1.645
B)1.645
C)18.84
D)45.16

Question 63

Suppose the life of a particular brand of laptop battery is normally distributed with a mean of 8 hours and a standard deviation of 0.6 hours.What is the probability that the battery will last more than 9 hours before running out of power?

A)0.0475
B)0.4525
C)0.9525
D)1.6667

Question 64

Exhibit 6-3.Patients scheduled to see their primary care physician at a particular hospital wait,on average,an additional eight minutes after their appointment is scheduled to start.Assume the time that patients wait is exponentially distributed. Refer to Exhibit 6-3.What is the probability a randomly selected patient will see the doctor within five minutes of the scheduled time?

A)0.2019
B)0.4647
C)0.5353
D)0.7981

Question 65

Find the mean of the lognormal variable if the mean and standard deviation of the underlying normal variable are 2 and 0.8,respectively.

A)0.69
B)2.32
C)10.18
D)11.02

Question 66

If an exponential distribution has the rate parameter λ = 5,what is its expected value?

A)5
B)1/5
C)1/25
D)5/2

Question 67

If has a lognormal distribution,what can be said of the distribution of the random variable X?

A)X follows a normal distribution.
B)X follows an exponential distribution.
C)X follows a standard normal distribution.
D)X follows a continuous uniform distribution.

Question 68

If an exponential distribution has the rate parameter λ = 5,what is its variance?

A)5
B)1/5
C)1/25
D)5/2

Question 69

Exhibit 6-4.The average time between trades for a high-frequency trading investment firm is 40 seconds.Assume the time between trades is exponentially distributed. Refer to Exhibit 6-4.What is the probability that the time between trades for a randomly selected trade and the one proceeding it is less than 20 seconds?

A)0.1354
B)0.3935
C)0.6065
D)0.8446

Question 70

Let the time between two consecutive arrivals at a grocery store check-out line be exponentially distributed with a mean of three minutes.Find the probability that the next arrival does not occur until at least four minutes have passed since the last arrival.

A)0.0000
B)0.2636
C)0.4724
D)0.7364

Question 71

Exhibit 6-2.Gold miners in Alaska have found,on average,12 ounces of gold per 1000 tons of dirt excavated with a standard deviation of 3 ounces.Assume the amount of gold found per 1000 tons of dirt is normally distributed. Refer to Exhibit 6-2.What is the probability the miners find between 10 and 14 ounces of gold in the next 1000 tons of dirt excavated?

A)0.2514
B)0.4972
C)0.5028
D)0.7486

Question 72

Exhibit 6-3.Patients scheduled to see their primary care physician at a particular hospital wait,on average,an additional eight minutes after their appointment is scheduled to start.Assume the time that patients wait is exponentially distributed. Refer to Exhibit 6-3.What is the probability a randomly selected patient will have to wait more than 10 minutes?

A)0.2865
B)0.4493
C)0.5507
D)0.7135

Question 73

Exhibit 6-4.The average time between trades for a high-frequency trading investment firm is 40 seconds.Assume the time between trades is exponentially distributed. Refer to Exhibit 6-4.What is the probability that the time between trades for a randomly selected trade and the one proceeding it is more than a minute?

A)0.2231
B)0.4869
C)0.5134
D)0.7769

Question 74

Find the variance of the lognormal variable if the mean and variance of the underlying normal variable are 2 and 1,respectively.

A)0
B)12.18
C)15.97
D)255.02

Question 75

Exhibit 6-5.The mean travel time to work is 25.2 minutes (U.S.Census 2010).Further,suppose that commute time follows a log-normal distribution with a standard deviation of 10 minutes. Refer to Exhibit 6-5.What is the probability a randomly selected U.S.worker has a commute time of more than half an hour?

A)25.78%
B)31.56%
C)68.44%
D)74.22%

Question 76

Exhibit 6-1.You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough.The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F. Refer to Exhibit 6-1.An inexpensive bag you are considering advertises to be good for temperatures down to 38°F.What is the probability that the bag will not be warm enough?

A)0.2266
B)0.2734
C)0.7500
D)0.7734

Question 77

Exhibit 6-2.Gold miners in Alaska have found,on average,12 ounces of gold per 1000 tons of dirt excavated with a standard deviation of 3 ounces.Assume the amount of gold found per 1000 tons of dirt is normally distributed. Refer to Exhibit 6-2.If the miners excavated 1000 tons of dirt,how little gold must they have found such that they find that amount or less only 15% of the time?

A)-1.04
B)1.04
C)8.88
D)15.12

Question 78

Exhibit 6-5.The mean travel time to work is 25.2 minutes (U.S.Census 2010).Further,suppose that commute time follows a log-normal distribution with a standard deviation of 10 minutes. Refer to Exhibit 6-5.What is the probability a randomly selected U.S.worker has a commute time of less than 20 minutes?

A)30.15%
B)34.09%
C)65.91%
D)69.85%

Question 79

What can be said about the expected value and standard deviation of an exponential distribution?

A)The expected value is equal to the standard deviation.
B)The expected value is equal to the square of the standard deviation.
C)The expected value is equal to the reciprocal of the standard deviation.
D)The expected value is equal to the square root of the standard deviation.

Question 80

Exhibit 6-1.You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough.The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F. Refer to Exhibit 6-1.One sleeping bag you are considering advertises that it is good for temperatures down to 25°F.What is the probability that this bag will be warm enough on a randomly selected May night at the park?

A)0.1894
B)0.3106
C)0.8106
D)0.8800