7. Consider the following scenario:

• Let P(C) = 0.2

• Let P(D) = 0.3

• Let P(C | D) = 0.4

• Part (a)
  Find P(C AND D).

• Part (b)
  Are C and D mutually exclusive? Why or why not?C and D are not mutually exclusive because
  P(C) + P(D) ≠ 1
  .C and D are mutually exclusive because they have different probabilities. C and D are not mutually exclusive because
  P(C AND D) ≠ 0
  .There is not enough information to determine if C and D are mutually exclusive.

• Part (c)
  Are C and D independent events? Why or why not?The events are not independent because the sum of the events is less than 1.The events are not independent because
  P(C) × P(D) ≠ P(C | D)
  . The events are not independent because
  P(C | D) ≠ P(C)
  .The events are independent because they are mutually exclusive.

• Part (d)
  Find P(D | C).

8. G and H are mutually exclusive events.

• P(G) = 0.5

• P(H) = 0.3

• Part (a)
  Explain why the following statement MUST be false:
  P(H | G) = 0.4.
  The events are mutually exclusive, which means they can be added together, and the sum is not 0.4.The statement is false because P(H | G) =

• = 0.6. To find conditional probability, divide
  P(G AND H) by P(H)
  , which gives 0.5.The events are mutually exclusive, which makes
  P(H AND G) = 0
  ; therefore,
  P(H | G) = 0.

• Part (b)
  Find
  P(H OR G).

• Part (c)
  Are G and H independent or dependent events? Explain

• G and H are dependent events because they are mutually exclusive.

• G and H are dependent events because

• P(G OR H) ≠ 1.

• G and H are independent events because they are mutually exclusive.

• There is not enough information to determine if G and H are independent or dependent events.

9.

Approximately 281,000,000 people over age five live in the United States. Of these people, 55,000,000 speak a language other than English at home. Of those who speak another language at home, 62.3 percent speak Spanish.

• E = speaks English at home

• E' = speaks another language at home

• S = speaks Spanish at home

Finish each probability statement by matching the correct answer.

• Part (a)
  P(E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (b)
  P(E)
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (c)
  P(S and E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (d)
  P(S | E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 29% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 4%.

Calculate the utility levels of each portfolio for an investor with A = 3. Assume the utility function is U = E(r) − 0.5 × Aσ². (Negative amounts should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to 4 decimal places.)

Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 37% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 5%.

Calculate the utility levels of each portfolio for an investor with A = 2. Assume the utility function is U = E(r) − 0.5 × Aσ². (Do not round intermediate calculations. Round your answers to 4 decimal places.)

Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 24% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 3%.

Calculate the utility levels of each portfolio for an investor with A = 2. Assume the utility function is U = E(r) − 0.5 × Aσ². (Do not round intermediate calculations. Round your answers to 4 decimal places.)

Table 1

Calculate the following for the car park using the data from Table 1:

a) the maximum real power demand P (kW)

b) maximum reactive power demand Q (kVAr)

c) the total apparent power demand S (kVA)

d) also, calculate the maximum current magnitude |I| if the supply voltage magnitude |V| is 3.3 kV and the supply frequency is 50 Hz.

e) Calculate the overall car park power factor.

f) Calculate the rating of the capacitor bank (μF) needed to improve the car park power factor to unity.

g) Calculate the new maximum current magnitude, after power factor correction. Again assume the supply voltage magnitude is 3.3kV and frequency is 50 Hz.

h) Select the cable ID from Table 2 that you would use to supply the car park BEFORE AND AFTER power factor correction has been applied.

i) Comment on your result.

Table 2

1. Five samples of table legs produced in an automated cutting process were taken each hour for 20 hours. The length of a table leg in centimeters was measured, and yielded averages and ranges show in the below table. Assume the sample size (n) is 5.

1. 1. Use the data to construct an chart and an R chart. Show all steps and all your work, including: Overall Mean and Overall Average Range; Calculations for Center Line, Upper Control Limit, and Lower Contol Limit, Visual representation of the control chart (either plotted in Excel or drawn to scale on graph paper). You must show all work for full credit.
   2. Is the process in control or out of control? Explain why thoroughly.
2. A fast-food franchise tracked the number of errors that occurred in customers’ orders. These included wrong menu item, wrong drink size, lack of condiments, wrong price total, etc. Some orders may have had more than one error. In one week, 1250 orders were filled, and a total of 30 errors were discovered. Find the Midline (, and the Upper and Lower Control Limits for a c chart. Show all your work for full credit.

Suppose you live in a world with only two risky assets: Amazon and Facebook (you can label these assets A and F, respectively). Furthermore, suppose these assets have the following expected returns and standard deviations:

a)If the correlation of Amazon and Facebook returns is equal to 0.3, what is the covariance of their returns?

Use Excel to calculate the expected return and standard deviation of the risky portfolio for different combinations of weights for A and F. In other words, set up your Excel spreadsheet to calculate the following:

etc… (for additional combinations of weights from w_A = -0.5 to w_A = 1.5)

Please include a screenshot of the filled-in table with your problem set solutions.

a)What is the optimal risky portfolio assuming r_f = 2% as in part e? Label this point P on your graph from part d.(Hint: Use Solver to choose the risky portfolio weights that maximize the Sharpe Ratio of the CAL created by combining the risky portfolio and the risk-free asset.)

b) Draw in the CAL through the optimal portfolio from part a.

c) What is the expected return and standard deviation of the complete portfolio that invests y=0.4 (i.e. 40%) in the optimal risky portfolio and 1-y = 0.6 (i.e 60%) in the risk-free asset.

d) If your utility from investing in the market is given by U = Er - 12σ² – 10σ⁴ , what is your optimal complete portfolio? (Hint: Use Solver to find the % of your complete portfolio in the risky asset (i.e. y) that maximizes your utility.)

e)What is a possible intuition of including the σ⁴ term in the utility function in part b?

27.58=0.5⋅(10^{(4.53678−(1149.360/(x+24.906))})+(0.3⋅(10^{(4.37576−(1175.581/(x−2.07)​)})+(0.2⋅(10^{(4.3281−(1132.108/(x+0.918)​)})

Hello We dont know how to solve for x in the above equation.