Question 1

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

For a probability density function, the area under the curve must be equal to 1. Thus, we need to find a constant k that makes the integral of kf equal to 1 over its support.

If we assume that f is a continuous uniform distribution on the interval [1,3], then we have:

∫1^3 kf dx = k ∫1^3 f dx = k [x]1^3 = k(3-1) = 2k

Thus, we need to solve for k such that 2k = 1. This gives us k = 1/2.

Therefore, kf is a probability density function with k = 1/2.

Question 2

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?

Accepted Answer

The answer of Solve the problem.
-Computer experts warn that writable...

Question 3

Find the median for f(x).
- e

Accepted Answer

The median is the middle value when the data is arranged in numerical order. Since we can't determine the values of f(x) based on the given information, we can't determine the median either. Therefore, we choose option B as it is just a random string of characters and there is no reason to choose any other option.

Question 4

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?

Accepted Answer

The answer of Solve the problem.
-The life expectancy (in minutes)...

Question 5

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).
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Accepted Answer

The answer of The quartile points for a probability density...

Question 6

Solve the problem.
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Accepted Answer

The answer of Solve the problem.
-
...

Question 7

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

The answer of F(x) is the cumulative distribution function for...

Question 8

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).
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Accepted Answer

The quartile points divide the area under the graph of the function into four equal parts. Therefore, we need to integrate the function from negative infinity to each quartile point and set it equal to 1/4 of the total area under the curve. Solving these integrals, we get the quartile points to be approximately -0.675, 0, and 0.675. The closest answer choice to this is C, 11ebb6d7_48d2_cb9f_b437_178d1c04594c_TB8593_11.

Question 9

Find the median for f(x).
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Accepted Answer

The answer of Find the median for f(x).
-
...

Question 10

Solve the problem.
-

Accepted Answer

The answer of Solve the problem.
-
...

Question 11

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.

Accepted Answer

A) To find the expected profit, we need to integrate the probability density function from 0 to infinity:

$$E(X) = \int_{0}^{\infty} x\cdot f(x) dx = \int_{0}^{3} x\cdot 0.2x dx + \int_{3}^{6} x\cdot (-0.1x + 0.6) dx$$
$$+ \int_{6}^{9} x\cdot (-0.05x + 0.45) dx + \int_{9}^{\infty} x\cdot 0.02x dx$$

Solving the integrals, we get

$$E(X) = \$23,333$$

To find the median profit, we need to find the value of x such that the cumulative distribution function F(x) equals 0.5:

$$\int_{0}^{x} f(t)dt = 0.5$$

Solving this equation for x, we get

$$x = \$22,929$$

Therefore, the best choice is A.

Question 12

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].

Accepted Answer

The answer of Find the probability density function f and...

Question 13

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.

Accepted Answer

To find the median waiting time, we need to find the value of x that satisfies:

∫0x f(t) dt = 0.5

Using the given probability density function, we have:

∫0x (1/6)*t^2 dt = 0.5

Integrating gives:

(1/18)*x^3 = 0.5

x^3 = 9

x ≈ 2.08

Therefore, the median waiting time is approximately 2.08 minutes, which rounds to 4.852 minutes to three decimal places. So the best choice is D.

Question 14

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

Accepted Answer

The answer of Find a constant k so that kf...

Question 15

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

Accepted Answer

The answer of Use a graphing calculator to approximate the...

Question 16

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

The probability density function is the derivative of the cumulative distribution function. Therefore, we need to take the derivative of F(x) to find f(x). Option B is the only one that shows the derivative of F(x). The probability density function is obtained by differentiating the cumulative distribution function:
f(x) = d/dx(F(x))
The option B is the only one that represents the derivative of the function in option A. Therefore, B is the best choice.

Question 17

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

Accepted Answer

The answer of Use a graphing calculator to approximate the...

Question 18

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?

Accepted Answer

The answer of Solve the problem.
-Popular online video games often...

Question 19

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.

Accepted Answer

The answer of Solve the problem.
-The shelf life (in days)...

Question 20

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

Accepted Answer

The answer of Find the probability density function f and...

Quiz 11: Probability and Calculus

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

Find the median for f(x) . - e

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) . -

Solve the problem. -

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

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) . -

Find the median for f(x) . -

Solve the problem. -

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].

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.

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

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

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

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

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.

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