Deck 11: Probability and Calculus

Question 1

Solve the problem.
Use the given function to find the indicated probabilities. Illustrate each probability with a graph.

Answer 1

Question 2

Find the associated cumulative distribution function. Graph both functions (on separate sets of axes).

Answer 2

Question 3

Find the associated cumulative distribution function. Graph both functions (on separate sets of axes).

A) -0.27
B) 0.23
C) 0.68
D) -0.31

Answer 3

0.68

Question 4

Find the value of the improper integral that converges.

A) 5
B)
C) The improper integral diverges.
D)

Question 5

Find the value of the improper integral that converges.

A) 9
B) 2
C) The improper integral diverges.
D)

Question 6

Solve the problem.
A trust fund produces a perpetual stream of income with rate flow Find the capital value at 5% compounded continuously.

A) $135,000
B) $54,000
C) $90,000
D) $33,750

Question 7

Graph y = f(x) and find the value of

A)
B)
C)
D)

Question 8

Solve the problem.
The perpetual annual rent for a property is $35,000. Find the capital value at 4% compounded continuously.

A) $98,500
B) $985,000
C) $875,000
D) $36,400

Question 9

Graph y = f(x) and find the value of

Question 10

Solve the problem.
Find and graph the cumulative distribution function associated with the function.

Question 11

Solve the problem.
Find and graph the cumulative distribution function associated with the function.

Question 12

Solve the problem.

A) (A) 4; (B) 4
B) (A) 2; (B) 6
C) (A) 1; (B) 3
D) (A) 2; (B) 4

Question 13

Find the value of the improper integral that converges.

A) The improper integral diverges.
B) -4
C) 0
D)

Question 14

Solve the problem.

A) (A) 25; (B) 6.25
B) (A) 2.5; (B) 1.125
C) (A) 5; (B) 2.5
D) (A) 20; (B) 80

Question 15

Find the associated cumulative distribution function. Graph both functions (on separate sets of axes).

A) -0.27
B) 0.38
C) -0.62
D) 0.40

Question 16

Graph y = f(x) and find the value of

Question 17

Solve the problem.
When a person takes a drug, the body does not assimilate all of the drug. One way to determine the amount of the drug assimilated is to measure the rate at which the drug is eliminated from the body. If the rate of
Elimination of the drug (in milliliters per minute) is given by where t is the time in
Minutes since the drug was administered, how much of the drug is eliminated from the body?

A) 10 mL
B) 2.5 mL
C) 25 mL
D) 100 mL

Question 18

Find F(b), use a graphing calculator to graph F and use the graph to estimate

A)
B)
C)
D)

Question 19

Find the associated cumulative distribution function. Graph both functions (on separate sets of axes).

Question 20

Solve the problem.
Use the given function to find the indicated probabilities. Illustrate each probability with a graph.

Question 21

Find a constant k so that kf is a probability density function, or state it does not exist.

A)
B)
C) k = 6
D) does not exist

Question 22

Solve the problem.
Computer experts warn that writable compact disks such as CD-Rs (Compact Disc-Recordable) and CD-RWs
(Compact Disc-ReWriteable) have much shorter lifespans than other digital storage methods such as hard
drives or magnetic tape. The time until the disc becomes unreadable for a particular brand of CD-RWs is a
continuous random variable with probability density function given by the following function. Round to two decimal
places. (A) What is the probability that a disc is not readable after 25 years?
(B) What is the probability that the disc is readable after 10 years?
(C) If the user wants the probability that the disc is readable to be 0.9, when is the last time it should be read?

Question 23

Find the median for f(x).
e

A)
B)
C)
D)

Question 24

Solve the problem.
The life expectancy (in minutes) of a certain microscopic organism is a continuous random variable with probability
density function given by the following function. Round to two decimal places. (A) What is the probability that an organism lives for at least 10 minutes?
(B) What is the probability that an organism lives for at most 8 minutes?

Question 25

The quartile points for a probability density function are the values that divide the area under the graph of the
function into four equal parts. Find the quartile points for the probability density function f(x).

A)
B)
C)
D)

Question 26

Solve the problem.

A)
B)
C)
D)

Question 27

F(x) is the cumulative distribution function for a continuous random variable X. Find the probability density function f(x)
associated with F(x).

A)
B)
C)
D)

Question 28

The quartile points for a probability density function are the values that divide the area under the graph of the
function into four equal parts. Find the quartile points for the probability density function f(x).

A)
B)
C)
D)

Question 29

Find the median for f(x).

A)
B)
C)
D)

Question 30

Solve the problem.

A) 48; 2,256; 47.497
B) 6; 48; 6.928
C) 96; 9,024; 94.995
D) 6; 12; 3.464

Question 31

Solve the problem.
A building contractor's profit (in thousands of dollars) on each unit in a subdivision is a continuous random variable with probability density function f(x) as shown below. (A) Find the contractor's expected profit. Round to the nearest dollar.
(B) Find the median profit. Round to the nearest dollar.

A) (A) $23,333; (B) $22,929
B) (A) $28,333; (B) $26,465
C) (A) $21,667; (B) $21,340
D) (A) $26,667; (B) $25,000

Question 32

Find the probability density function f and the associated cumulative distribution function F for the continuous random
variable X if
X is uniformly distributed on [1, 7].

A)
B) 7
C)
D)

Question 33

Solve the problem.
The time (in minutes) a customer must wait in line at a bank is a continuous random variable with probability density function f(x) as shown below. Find the median waiting time. Round to three decimal places.

A) 0.693 minutes
B) 0.143 minutes
C) 3.892 minutes
D) 4.852 minutes

Question 34

Find a constant k so that kf is a probability density function, or state it does not exist.

A)
B) k = -2
C) does not exist
D) k = 2

Question 35

Use a graphing calculator to approximate the median of the indicated probability density function f. Round to two
decimal places.

A) 1.78
B) 1.27
C) 1.56
D) 1.31

Question 36

F(x) is the cumulative distribution function for a continuous random variable X. Find the probability density function f(x)
associated with F(x).

A)
B)
C)
D)

Question 37

Use a graphing calculator to approximate the median of the indicated probability density function f. Round to two
decimal places.

A) 0.46
B) 0.36
C) 0.63
D) 0.66

Question 38

Solve the problem.
Popular online video games often struggle to meet the initially high demand for access to the game servers. It is
costly to increase server capacity and the demand always falls after a few months, but if the wait times are too
long the game could lose sales as a result of negative criticism. The time (in minutes) a customer must wait to
get into the game is a continuous random variable with probability density function given by the following
function. Round to two decimal places. (A) Evaluate dx and interpret the results.
(B) What is the probability that a customer waits less than 7 minutes?
(C) What is the probability that a customer waits more than 10 minutes?

Question 39

Solve the problem.
The shelf life (in days) of a perishable drug is a continuous random variable with probability density function f(x) as shown below. Find the median shelf life.

A) 125 days
B) 5 days
C) 15 days
D) 25 days

Question 40

Find the probability density function f and the associated cumulative distribution function F for the continuous random
variable X if
X is an exponential random variable with

A)
B)
C)
D)

Question 41

Solve the problem.
Find the area under the standard normal curve from 0 to the indicated value of z. 0.85

A) 0.3023
B) 0.8023
C) 0.6977
D) 0.1977

Question 42

X is a continuous random variable with mean and then find ) if
X is an exponential random variable with λ = 4.

A)
B)
C)
D)

Question 43

Find the mean, median, and standard deviation of the continuous random variable X if
X is uniformly distributed on [4, 8].

A)
B)
C)
D)

Question 44

Given a normal distribution with mean 40 and standard deviation 8, find the number of standard deviations the
measurement is from the mean. Express the answer as a positive number.
24

A) 8
B) 16
C) 0.5
D) 2

Question 45

Solve the problem.
The time (in minutes) applicants must wait to receive a driver's examination is uniformly distributed on the interval [0, 80]. What is the probability that an applicant must wait more than 30 minutes?

A) 0.8125
B) 0.375
C) 0.1875
D) 0.625

Question 46

Given a normal distribution with mean 150 and standard deviation 25, find the indicated probability.

A) 0.8415
B) 0.1585
C) 0.5793
D) 0.4207

Question 47

Solve the problem.
The life expectancy of a car battery is normally distributed. The average (mean) lifetime is 235 weeks with a standard deviation of 15 weeks. If the company guarantees the battery for 4 years, what percentage of the
Batteries sold would be expected to be returned before the end of the warranty period?

A) 0.0719
B) 0.9281
C) 0.0359
D) 0.9641

Question 48

Solve the problem.
Find the area under the standard normal curve and above the given interval on the horizontal axis.

A) 0.0557
B) 0.4236
C) 0.4793
D) 0.5207

Question 49

Given a normal distribution with mean 40 and standard deviation 8, find the number of standard deviations the
measurement is from the mean. Express the answer as a positive number.
50

A) 1.25
B) 0.8
C) 10
D) 2

Question 50

Find the mean, median, and standard deviation of the continuous random variable X if
X is an exponential random variable with λ = 2.

A)
B)
C)
D)

Question 51

Solve the problem.
The length of time for telephone conversations (in minutes) is exponentially distributed. The average (mean) length of a conversation is 5 minutes. What is the probability that a conversation lasts less than 3 minutes?

A) 0.189
B) 0.451
C) 0.549
D) 0.811

Question 52

Given a normal distribution with mean -10 and standard deviation 5, find the area under the normal curve and above the
given interval on the horizontal axis.
(-∞ , -5]

A) 0.6827
B) 0.1587
C) 0.3173
D) 0.8413

Question 53

X is a continuous random variable with mean µ and standard deviation σ. Find µ and σ, and then find P(µ -σ ≤ X ≤ µ +σ)
if
X is uniformly distributed on [-6, 6].

A)
B)
C)
D)

Question 54

Given a normal distribution with mean -10 and standard deviation 5, find the area under the normal curve and above the
given interval on the horizontal axis.

A) 0.0228
B) 0.1815
C) 0.1587
D) 0.1359

Question 55

X is a continuous random variable with mean µ and standard deviation σ. Find µ and σ, and then find P(µ -σ ≤ X ≤ µ +σ)
if
X is an exponential random variable with m = 2 ln 2.

A)
B)
C)
D)

Question 56

X is a continuous random variable with mean and then find ) if
X is uniformly distributed on [0, 6].

A)
B)
C)
D)

Question 57

Given a normal distribution with mean 150 and standard deviation 25, find the indicated probability.

A) 0.8849
B) 0.2118
C) 0.3269
D) 0.6731