Question 1

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

Accepted Answer

The probability that the average time interval between arrivals of successive cars is more than 17 seconds is given by the cumulative distribution function of the exponential distribution:```P(X > 17) = 1 - P(X <= 17) = 1 - (1 - e^(-17/8)) = e^(-17/8) = 0.12```

Question 2

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)

Accepted Answer

The answer of The life span (in years) of a...

Question 3

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

Accepted Answer

The answer of The quantity demanded x (in thousands of...

Question 4

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

Accepted Answer

The answer of Determine whether the given function is a...

Question 5

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

Accepted Answer

The answer of The amount of snowfall in feet in...

Question 6

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

Accepted Answer

The answer of The quantity demanded x (in thousands of...

Question 7

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

Accepted Answer

The answer of The life span (in years) of a...

Question 8

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

Accepted Answer

The answer of Determine whether the given function is a...

Question 9

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

Accepted Answer

The probability that a microprocessor will fail during the warranty period is given by the cumulative distribution function of the exponential distribution:P(X < 20,000) = 1 - e^(-20,000/120,000) = 0.15

Question 10

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.

Accepted Answer

The answer of The microprocessors, which are used in automobiles...

Question 11

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

Accepted Answer

A. Using the exponential distribution, we can calculate the probability of a breakdown between 300 and 800 hours as follows:
P(300 < X < 800) = e^(-300/1000) - e^(-800/1000) = 0.29
B. The probability of a robot breaking down after 1,200 hours is simply the probability that it lasts longer than 1,200 hours, which is:
P(X > 1200) = e^(-1200/1000) = 0.3

Question 12

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

Accepted Answer

The answer of Determine whether the given function is a...

Question 13

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

Accepted Answer

The answer of Determine whether the given function is a...

Question 14

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

Accepted Answer

The answer of Determine whether the statement is true or...

Question 15

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.

Accepted Answer

The answer of The number of chocolate chips in each...

Question 16

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.

Accepted Answer

The answer of Suppose the time intervals between arrivals of...

Question 17

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)

Accepted Answer

We need to find the probability that at most half of the streets need repairs, which means we need to find the probability that X ≤ 50 (where X is the number of streets needing repairs). To do this, we need to integrate the given probability density function from 0 to 50:

P(X ≤ 50) = ∫0^50 0.00022x^2 e^(-0.0011x) dx

This integral can be evaluated using integration by parts, or by using a table of integrals. Either way, we get:

P(X ≤ 50) ≈ 0.6604

So the answer is C, 11eaa8b1_da03_7a2c_b1b6_29f115c46b11_TB6026_11.

Question 18

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

Accepted Answer

The answer of Determine whether the given function is a...

Question 19

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

Accepted Answer

The answer of The number of chocolate chips in each...

Question 20

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

Accepted Answer

The answer of Determine whether the given function is a...

Quiz 7: Additional Topics in Integration

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)

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?

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

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

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

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?

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

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

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

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

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.

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.

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

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.

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