Deck 7: Additional Topics in Integration

Question 1

Determine whether the given function is a probability density function on the specified interval.

A) yes
B) no

Answer 1

no

Question 2

Determine whether the given function is a probability density function on the specified interval.

A) no
B) yes

Answer 2

yes

Question 3

What function is a probability density function on the interval ?

A)
B)
C)
D)
E)

Answer 3

Question 4

Determine the value of the constant k so that the function is a probability density function on the interval .

A)
B)
C)
D)
E)

Question 5

Determine the value of the constant so that the function is a probability density function on the interval . If is a continuous random variable, compute the probability that will assume a value between and .

A)
B)
C)
D)
E)

Question 6

Determine whether the given function is a probability density function on the specified interval.

A) yes
B) no

Question 7

Determine whether the given function is a probability density function on the specified interval.

A) no
B) yes

Question 8

Determine the value of the constant k so that the function is a probability density function on the interval .

A)
B)
C)
D)
E)

Question 9

Find the expected value of the continuous random variable X associated with the probability density function over the indicated interval.

A) 1.49
B) 1.3333
C) 2.772
D) 1.386

Question 10

Find the expected value of the continuous random variable X associated with the probability density function over the indicated interval.

A)
B)
C)
D)

Question 11

Determine the value of the constant k so that the function is a probability density function on the interval .

A)
B)
C)
D)

Question 12

Determine the value of the constant k so that the function is a probability density function on the interval .

A)
B)
C)
D)

Question 13

Find the expected value of the continuous random variable X associated with the probability density function over the indicated interval.

A)
B)
C)
D)
E)

Question 14

The amount of time t (in seconds) it takes a motorist to react to a road emergency is a continuous random variable with probability density function What is the expected reaction time for a motorist chosen at random?

A) 1.86 sec
B) 1.88 sec
C) 1.55 sec
D) 1.5 sec

Question 15

Determine whether the given function is a probability density function on the specified interval.

A) no
B) yes

Question 16

The average waiting time for patients arriving at the Newtown Health Clinic between 1 P.M. and 4 P.M. on a weekday is an exponentially distributed random variable x with expected value of 30 minutes. What is the probability that a patient arriving at the clinic between 1 P.M. and 4 P.M. will have to wait between 27 and 32 minutes?
Round your answer to two decimal places.

A) 0.31
B) 0.16
C) 0.17
D) 0.06

Question 17

Determine whether the given function is a probability density function on the specified interval.

A) no
B) yes

Question 18

The amount of snowfall in feet in a remote region of Alaska in the month of January is a continuous random variable with probability density function Find the amount of snowfall one can expect in any given month of January in Alaska.

A) 7.5 ft
B) 4 ft
C) 2.5 ft
D) 3.5 ft

Question 19

Determine whether the given function is a probability density function on the specified interval.

A) yes
B) no

Question 20

The life span of a certain plant species (in days) is described by the probability density function . Find the probability that a plant of this species will live more than 120 days.

A) 0.736
B) 0.643
C) 0.423
D) 0.264
E) 0.569

Question 21

Suppose the time intervals between arrivals of successive cars at an expressway tollbooth during rush hour are exponentially distributed and that the average time interval between arrivals is 8 seconds. Find the probability that the average time interval between arrivals of successive cars is more than 17 seconds. Round your answer to two decimal places.

A) 0.12
B) 0.64
C) 0.62
D) 0.23

Question 22

The life span (in years) of a certain brand of color television tube is a continuous random variable with probability density function How long is one of these color television tubes expected to last?
__________ year(s)

Question 23

The quantity demanded x (in thousands of pounds) of a certain brand of butter each week is a continuous random variable with probability density function What is the expected demand for this brand of butter each week?

A) 1,000 lb
B) 4,000 lb
C) 8,000 lb
D) 9,000 lb

Question 24

Determine whether the given function is a probability density function on the specified interval.

Question 25

The amount of snowfall in feet in a remote region of Alaska in the month of January is a continuous random variable with probability density function Find the amount of snowfall one can expect in any given month of January in Alaska.
__________ ft

Question 26

The quantity demanded x (in thousands of pounds) of a certain brand of butter each week is a continuous random variable with probability density function What is the expected demand for this brand of butter each week?
__________ lb

Question 27

The life span (in years) of a certain brand of color television tube is a continuous random variable with probability density function How long is one of these color television tubes expected to last?

A) 13 years
B) 5 years
C) 11 years
D) 15 years
E) 3 years

Question 28

Determine whether the given function is a probability density function on the specified interval.

Question 29

The microprocessors, which are used in automobiles to regulate fuel consumption, are guaranteed against defects for 20,000 miles of use. Tests conducted in the laboratory under simulated driving conditions reveal that the distances driven before the microprocessors break down are exponentially distributed and that the average distance driven before the microprocessors fail is 120,000 miles. What is the probability that a microprocessor selected at random will fail during the warranty period?

A) 0.15
B) 0.12
C) 0.36
D) 0.63
E) 0.85

Question 30

The microprocessors, which are used in automobiles to regulate fuel consumption, are guaranteed against defects for 15,000 miles of use. Tests conducted in the laboratory under simulated driving conditions reveal that the distances driven before the microprocessors break down are exponentially distributed and that the average distance driven before the microprocessors fail is 80,000 miles. What is the probability that a microprocessor selected at random will fail during the warranty period? Please round the answer to the nearest hundredth.

Question 31

National Welding uses industrial robots in some of its assembly-line operations. Management has determined that, on average, a robot breaks down after 1,000 hours of use and that the lengths of time between breakdowns are exponentially distributed.
A. What is the probability that a robot selected at random will break down between 300 and 800 hours of use?
Round your answer to two decimal places.
B. What is the probability that a robot will break down after 1,200 hours of use?
Round your answer to two decimal places.

A) a. 0.29, b. 0.3
B) a. 0.61, b. 0.13
C) a. 0.76, b. 0.44
D) a. 0.26, b. 0.61

Question 32

Determine whether the given function is a probability density function on the specified interval.

Question 33

Determine whether the given function is a probability density function on the specified interval.

Question 34

Determine whether the statement is true or false. If is a probability density function of a continuous random variable in the interval , then the expected value of is given by

A) true
B) false

Question 35

The number of chocolate chips in each cookie of a certain brand has a distribution described by the probability density function Find the expected number of chips in each cookie.

Question 36

Suppose the time intervals between arrivals of successive cars at an expressway tollbooth during rush hour are exponentially distributed and that the average time interval between arrivals is 8 seconds. Find the probability that the average time interval between arrivals of successive cars is more than 8 seconds.
Round your answer to two decimal places.

Question 37

The number of streets in the downtown section of a certain city that need repairs in a given year is a random variable with a distribution described by the probability density function Find the probability that at most half of the streets will need repairs in any given year.

A)
B)
C)
D)
E)

Question 38

Determine whether the given function is a probability density function on the specified interval.

Question 39

The number of chocolate chips in each cookie of a certain brand has a distribution described by the probability density function Find the expected number of chips in each cookie.

A) 7
B) 3
C) 4
D) 2

Question 40

Determine whether the given function is a probability density function on the specified interval.

Question 41

Find the area of the region under the given curve over the indicated interval.

A)
B)
C)
D)
E)

Question 42

The life span of a certain plant species (in days) is described by the probability density function Find the probability that a plant of this species will live for 95 days or less. Please round the answer to the nearest thousandth if necessary. __________
Find the probability that a plant of this species will live more than 130 days. Please round the answer to the nearest thousandth if necessary. __________
Find the probability that a plant of this species will live more than 60 days but less than 150 days. Please round the answer to the nearest thousandth if necessary. __________

Question 43

The amount of time t (in seconds) it takes a motorist to react to a road emergency is a continuous random variable with probability density function What is the expected reaction time for a motorist chosen at random?

Question 44

Determine whether the statement is true or false.
If exists, then exists and .

Question 45

Find the area of the region under the given curve over the indicated interval.

A)
B)
C)
D)
E)

Question 46

Evaluate the following improper integral whenever it is convergent.

A) 5
B) 6
C) 7
D) 9
E) the integral is divergent

Question 47

Show that the function is a probability density function on the interval .

Question 48

Find the area of the region under the given curve over the indicated interval.

A) sq unit
B) sq units
C) sq unit
D) sq unit
E) sq unit

Question 49

Find the area of the region under the given curve over the indicated interval.

A) 10 sq units
B) 12 sq units
C) 9 sq units
D) 6 sq units

Question 50

Find the area of the region bounded by the x-axis and the graph of the function.

A) sq units
B) sq units
C) sq units
D) sq units
E) sq units

Question 51

Determine whether the given function is a probability density function on the specified interval.

Question 52

Find the area of the region under the given curve over the indicated interval.

A) sq unit
B) sq unit
C) sq unit
D) sq units
E) sq unit

Question 53

The number of streets in the downtown section of a certain city that need repairs in a given year is a random variable with a distribution described by the probability density function Find the probability that at most half of the streets will need repairs in any given year.

Question 54

Find the area of the region under the curve over the indicated interval.

A) sq units
B) sq units
C) sq units
D) sq units
E) sq units

Question 55

Determine whether the statement is true or false.
If is a probability density function of a continuous random variable in the interval , then the expected value of is given by

Question 56

Find the area of the region bounded by the x-axis and the graph of the function

A) 4 sq units
B) sq unit
C) sq unit
D) 0 sq units

Question 57

Evaluate the following improper integral whenever it is convergent.

A) 17
B) 26
C) 8
D) 18
E) the integral is divergent

Question 58

a. Determine the value of the constant k so that the function is a probability density function on the interval . __________
b. If x is a continuous random variable with the probability density function given in part (a), find the probability that x will assume a value greater than 1.
Round your answer to four decimal places.

Question 59

Find the area of the region under the given curve over the indicated interval.

A) sq units
B) sq units
C) sq units
D) sq units

Question 60

National Welding uses industrial robots in some of its assembly-line operations. Management has determined that, on average, a robot breaks down after 1,000 hours of use and that the lengths of time between breakdowns are exponentially distributed.
a. What is the probability that a robot selected at random will break down between 300 and 600 hours of use?
Round your answer to two decimal places.
b. What is the probability that a robot will break down after 1,100 hours of use?
Round your answer to two decimal places.

Question 61

A university alumni group wishes to provide an annual scholarship in the amount of $1,800 beginning next year. If the scholarship fund will earn an interest rate of 9% / year compounded continuously, find the amount of the endowment the alumni are required to make now.

A) $20,473
B) $20,000
C) $19,672
D) $19,017

Question 62

Find the area of the region bounded by the x-axis and the graph of the function. __________ sq units

Question 63

Evaluate the following improper integral whenever it is convergent.

A)
B)
C) 4
D) - 4
E) the integral is divergent

Question 64

Evaluate the following improper integral whenever it is convergent.

Question 65

Find the area of the region under the curve over the indicated interval. __________ sq units

Question 66

Evaluate the following improper integral whenever it is convergent.

A)
B)
C)
D)
E) the integral is divergent

Question 67

Evaluate the following improper integral whenever it is convergent.

A)
B)
C)
D)
E) the integral is divergent

Question 68

Evaluate the improper integral whenever it is convergent.

A)
B)
C)
D)
E) divergent

Question 69

Find the area of the region under the given curve over the indicated interval.

Question 70

Evaluate the following improper integral whenever it is convergent.

Question 71

Evaluate the following improper integral whenever it is convergent.

A) 2
B) -5
C) 1
D) -6
E) the integral is divergent

Question 72

The present value of a perpetual stream of income that flows continually at the rate of dollars per year is given by the formula where is the interest rate compounded continuously. Using this formula, find the present value of a perpetual net income stream that is generated at the rate of dollars per year. Hint: .

A)
B)
C)
D)
E)

Question 73

Evaluate the following improper integral whenever it is convergent.

A) 8
B) -2
C) 5
D) 0
E) the integral is divergent

Question 74

Evaluate the improper integral if it is convergent.

A) 3
B) -3
C) 2
D) 4
E) divergent

Question 75

Evaluate the following improper integral whenever it is convergent.

A)
B)
C)
D)
E) the integral is divergent

Question 76

Evaluate the following improper integral whenever it is convergent.

A)
B)
C)
D)
E)

Question 77

Evaluate the following improper integral whenever it is convergent.

A) 4
B) -1
C) 1
D) 7
E) the integral is divergent

Question 78

Evaluate the following improper integral whenever it is convergent.

A)
B) 12
C)
D)
E) the integral is divergent

Question 79

Evaluate the following improper integral whenever it is convergent.

Question 80

Becky Wilkinson wants to establish a trust fund that will provide her children and heirs with a perpetual annuity in the amount of thousand dollars per year beginning next year. If the trust fund will earn an interest rate of 10% per year compounded continuously, find the amount that she must place in the trust fund now. Hint: Use the formula .

A)
B)
C)
D)
E)