1.(10 points) Assume that the Hospital Fatality Rate (HFR) for patients hospitalized due to COVID-19 is 20%.This means that of all patients who test positive for COVID-19 and are hospitalized due to the severity of their symptoms, 20% die rather than recover.If a hospital currently has 12 confirmed COVID-19 patients, what is the probability that at least 10 of them will recover and survive?

2.(10 points) A portfolio has an expected monthly return of 3% with a standard deviation of 6%.

a.What is the probability that this portfolio will have a negative return in one month?

b.What is the probability that this portfolio will have a return greater than 10% in a month?

c.Over one year, what is the probability that the average of the monthly returns for this portfolio will be negative?

3.(3 points)Assume that on weekdays after the morning rush hour but before the evening rush hour the wait time for the CTA Red Line subway is continuously uniformly distributed between 0 and 8 minutes.

a.What is the expected wait time on a weekday around mid-day?

b.What is the probability of waiting for two minutes or less on a weekday around mid-day?

c.What is the probability of waiting for two minutes or less around mid-dayon three weekdays in a row?

4.(3 points)Over the past 3 years (n = 36 months), the average monthly returns for Ford Motor Company stock were 0.025% with a standard deviation of 7.098%.Construct the 95% confidence interval for the mean value of the monthly return on Ford Motor Company stock

5.(7 points)A client wants to make a year end charitable donation of $10,000 from her portfolio of stocks and bonds which is worth $400,000 on June 30, but she wants to be able to make the donation without invading the current principle of $400,000.You estimate from her current allocations that the mean expected return on her portfolio over six months will be 15.7% with a standard deviation of 14.5%.You consider two possible reallocations.The more aggressive reallocation would have an expected return of 19.8% with a standard deviation of 19.8%, while the more conservative reallocation would have an expected return of 12.6% with a standard deviation of 10.3%.

a.Given her desire to make the donation without invading her $400,000 principle, what is the shortfall level to use for the safety-first rule?

b.According to the safety-first criterion, should you recommend the more aggressive reallocation, the more conservative reallocation, or the current allocation?

c.For the portfolio that is optimal according to the safety-first criterion, what is the probability that the return will be less than the shortfall level?