Two students in a Game Theory course plan to take an exam tomorrow. The professor seeks to create incentives for students to study, so he tells them that the student with the highest score will receive a grade of A and the one with the lower score will receive a B. Student 1’s score equals x1 + 1.5, where x1 denotes the amount of hours studying. Student 2’s score equals x2, where x2 is the hours she studies. Note that these score functions imply that, if both students study the same number of hours, x1 = x2, student 1 obtains the higher score, i.e., she is “the smarter of the two”. Assume, for simplicity, that the hours of studying for the game theory course is an integer number, and that they cannot exceed 5. The payoff to student i is 10 – xi if she gets an A and 8 – xi if she gets a B.

a) Which outcome(s) survive iterated deletion of strictly dominated strategies?

b) Which outcomes survive iterated deletion of weakly dominated strategies?

Write a program that uses a custom function to generate a specified number of random integers in a specified range. This custom function should take three arguments; the number of integers to generate, the lower limit for the range, and the upper limit for the range. Values for these arguments should be entered by the user in main. The custom function should display the random integers on one line separated by single spaces. The function should also report how many numbers were even and how many were odd. Finally, the function should calculate the total of the random integers and return this total back to main where it will be printed. User inputs in main are shown in blue.
Sample Output
How many integers are to be generated 8
Enter the lowest integer desired 10
Enter the highest integer desired 20
16 14 19 20 12 15 18 18
6 evens and 2 odds were generated
The total of those 8 random numbers is 132

1. Consider the probability distribution shown below.

Compute the expected value of the distribution. (Enter a number.)


Compute the standard deviation of the distribution. (Enter a number. Round your answer to four decimal places.)

2. What is the income distribution of super shoppers? A supermarket super shopper is defined as a shopper for whom at least 70% of the items purchased were on sale or purchased with a coupon. In the following table, income units are in thousands of dollars, and each interval goes up to but does not include the given high value. The midpoints are given to the nearest thousand dollars.

(a)

Using the income midpoints x and the percent of super shoppers, do we have a valid probability distribution? Explain.

Yes. The events are distinct and the probabilities sum to 1.

Yes. The events are indistinct and the probabilities sum to less than 1.  

No. The events are indistinct and the probabilities sum to 1.

No. The events are indistinct and the probabilities sum to more than 1.

Yes. The events are distinct and the probabilities do not sum to 1.

(b)

Use a histogram to graph the probability distribution of part (a). (Select the correct graph.)

A.

B.    

C.

D.

(c)Compute the expected income μ of a super shopper (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)
μ =  thousands of dollars

(d) Compute the standard deviation σ for the income of super shoppers (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)
σ =  thousands of dollars

A simple pendulum with mass m = 1.8 kg and length L = 2.69 m hangs from the ceiling. It is pulled back to an small angle of θ = 8.7° from the vertical and released at t = 0.

1) What is the period of oscillation?

2) What is the magnitude of the force on the pendulum bob perpendicular to the string at t=0?

3) What is the maximum speed of the pendulum?

4) What is the angular displacement at t = 3.65 s? (give the answer as a negative angle if the angle is to the left of the vertical)

5) What is the magnitude of the tangential acceleration as the pendulum passes through the equilibrium position?

6) What is the magnitude of the radial acceleration as the pendulum passes through the equilibrium position?

7) Which of the following would change the frequency of oscillation of this simple pendulum?

increasing the mass

decreasing the initial angular displacement

increasing the length

hanging the pendulum in an elevator accelerating downward

The table below shows the number of deaths in the U.S. in a year due to a variety of causes. For these questions, assume these values are not changing from year to year, and that the population of the United States is 312 million people. Cause Deaths Passenger car occupant (driver or rider) 13,100 Motorcycle (driver or rider) 4,500 Tornado 553 Skydiving 56

a) What is the probability that an American chosen at random died as a passenger car occupant last year? 0.719 Incorrect Give your answer as a fraction or decimal. If decimal, make sure your answer is accurate to at least 2 significant figures (values after leading zeros)

The table below shows the number of deaths in the U.S. in a year due to a variety of causes. For these questions, assume these values are not changing from year to year, and that the population of the United States is 312 million people. Cause Deaths Passenger car occupant (driver or rider) 13,100 Motorcycle (driver or rider) 4,500 Tornado 553 Skydiving 56 Hint: b)Does your probability of dying in a car accident next year differ much from the probability of a random person dying in a car accident?

c) Estimate the probability that you will die as a passenger car occupant next year? Incorrect 0.0000018 Make sure your answer is accurate to at least 2 significant figures (values after leading zeros)

The table below shows the number of deaths in the U.S. in a year due to a variety of causes. For these questions, assume these values are not changing from year to year, and that the population of the United States is 312 million people. Cause Deaths Passenger car occupant (driver or rider) 13,100 Motorcycle (driver or rider) 4,500 Tornado 553 Skydiving 56

d) What is the probability that an American chosen at random will die as the result of a tornado next year? 0.25Incorrect Make sure your answer is accurate to at least 2 significant figures (values after leading zeros)

The table below shows the number of deaths in the U.S. in a year due to a variety of causes. For these questions, assume these values are not changing. Cause Deaths Passenger car occupant (driver or rider) 13,100 Motorcycle (driver or rider) 4,500 Tornado 553 Skydiving 56

g) People sometimes claim skydiving is less dangerous than driving or riding in a car. Does the data support this claim? Explain. The table below shows the number of deaths in the U.S. in a year due to a variety of causes. For these questions, assume these values are not changing. Cause Deaths Passenger car occupant (driver or rider) 13,100 Motorcycle (driver or rider) 4,500 Tornado 553 Skydiving 56

h) People sometimes claim motorcycle riding is less dangerous than traveling by car. Does the data support this claim? What additional information and/or calculations would be useful to evaluate this claim?

It is known from past information that the probability of failing to finish an Ironman (called a DNF) is 15%. Suppose we take a sample of 100 Ironman races over the past few years and find that the DNF percentage is 12%. We are interested in seeing if there has been a significant decrease in the number of DNFs. Use alpha=0.05. Find the 95% confidence interval for the percentage of DNFs.

It is known from past information that the probability of failing to finish an Ironman (called a DNF) is 15%. Suppose we take a sample of 100 Ironman races over the past few years and find that the DNF percentage is 12%. We are interested in seeing if there has been a significant decrease in the number of DNFs. Use alpha=0.05. The null and the alternative hypotheses are:

It is known from past information that the probability of failing to finish an Ironman (called a DNF) is 15%. Suppose we take a sample of 100 Ironman races over the past few years and find that the DNF percentage is 12%. We are interested in seeing if there has been a significant decrease in the number of DNFs. Use alpha=0.05. The test statistic is:

It is known from past information that the probability of failing to finish an Ironman (called a DNF) is 15%. Suppose we take a sample of 100 Ironman races over the past few years and find that the DNF percentage is 12%. We are interested in seeing if there has been a significant decrease in the number of DNFs. Use alpha=0.05. The p-value is:

It is known from past information that the probability of failing to finish an Ironman (called a DNF) is 15%. Suppose we take a sample of 100 Ironman races over the past few years and find that the DNF percentage is 12%. We are interested in seeing if there has been a significant decrease in the number of DNFs. Use alpha=0.05. Assuming no prior estimate of p is available, what is the sample size that would need to be taken if we wanted to have a margin of error of 0.08 or less with 95% confidence?

CHAPTER 3: RADIOLOGY Case 3-1 Interventional radiologists perform both the procedure and the radiologic services, so you will report both the procedure and radiology services. LOCATION: Outpatient, Hospital PATIENT: Mike Morgan PRIMARY CARE PHYSICIAN: Ronald Green, MD INTERVENTIONAL RADIOLOGIST: Edward Riddle, MD EXAMINATION: CT-guided liver biopsy. CLINICAL SYMPTOMS: Liver mass. CT-GUIDED LIVER BIOPSY: Informed consent was obtained. The patient was placed supine on the CT table, and axial CT was performed to localize the low-density lesion within the dome of the liver medially. The right mid-axillary line skin was prepped and draped in the usual sterile fashion. The skin and subcutaneous tissues were infiltrated with 1% lidocaine. A 19-gauge coaxial needle was advanced into the low-density lesion, and axial CT was performed to confirm needle position prior to biopsy. Four 20-gauge core biopsy samples were obtained. Biopsy samples were obtained using a 20-gauge Monotype biopsy gun. The biopsy gun and coaxial needle were removed. The patient did not receive conscious sedation. His pulse oximeter and vital signs were monitored throughout the exam. There were no complications. He tolerated the procedure well and left the radiology department in stable condition. IMPRESSION: Successful and uncomplicated CT-guided biopsy of a low-density mass within the dome of the liver medially. Pathology Report Later Indicated: Primary malignancy of the liver. CPT Code(s): _____________ ICD-10-CM Code(s): ___________ Abstracting Questions: 1. What technique was used to accomplish the liver biopsy? 2. What type of radiological guidance was used? 3. Is the radiological guidance reported separately? _ 4. Was sedation administered?

A closed piston-cylinder system undergoes a cycle:
​Process 1 to 2 is an expansion process where heat is added. The temperature remains constant.
Process 2 to 3 is a constant volume heat addition process.
​Process 3 to 4 is a compression process where heat is lost. The pressure remains constant.
​Process 4 to 1 is polytropic expansion, Pvⁿ​=constant with n=1.4 and it is adiabatic (Q=0).
​Assume air, ideal gas, with constant specific heats, k=1.4. R=.287kJ/kgK
​State 1: T=300K, P=150kPa
​State 2: T=300K, P=100kPa
State 3: P=600kPa
​State 4: P=600kPa

​a.) Find the temperature for State 3 and State 4.
​b.) For all 4 States, find volume(m³/kg)
​c.) For each process, (1-2, 2-3, 3-4, 4-1), find Q/m(kJ/kg), W/m(kJ/kg), and Δu(kJ/kg)
​d.) Plot all 4 States on a P-v diagram
​e.) Is this a power or refrigeration cycle?
​f.) Calculate the efficiency or coefficient of performance based on your answer to part e.

A gas confined in a piston-cylinder arrangement occupies 0.5 liters (Vi) at 4 atm (Pi) and 278 K (Ti). From this initial state the gas is allowed to expand against the surrounding atmosphere which is at a constant pressure of 1 atm (Patm) and a temperature of 278 K. At equilibrium, the gas’s final volume (Vf) is 2 liters and its final temperature (Tf) is 278 K. The initial and final states are indicated in the figure below.

a. In going from its initial to final state, did the gas do work on the atmosphere or did the atmosphere do work on the gas?

b. Calculate the work (w) and report t with the appropriate sign according to the convention presented in the text

c. If the surroundings transferred 100 joules of thermal energy to the system in going from its initial state to its final state, what is the change in internal energy (∆U) of the gas

d. Is the system open, closed, or isolated

e. What is the change in energy of the surroundings (∆Usurr)

An insulated cylinder fitted with a movable piston to maintain constant pressure initially contains 100 g of ice at -10 C. Heat is supplied to the contents at a constant rate by a 100 W heater. Make a graph showing temperature of the cylinder contents as a function of time starting at t = 0, when the temperature is -10 C and ending when the temperature is 110 C. Let c = 2.0 kJ/kg K for specific heat of ice from -10 to 0 C and of the steam from 100 to 110 C. The specific heat of water between 0 C and 100 C is c = 4.18 X 103 J/kg K. The value of Lv = 2257 X 103 J/ kg and Lf = 333.5 X 103 J/kg