A city has just added 110 new female recruits to its police force. The city will provide a pension to each new hire who remains with the force until retirement. In addition, if the new hire is married at the time of her retirement, a second pension will be provided for her husband. A consulting actuary makes the following assumptions: (i) Each new recruit has a 0.3 probability of remaining with the police force until retirement. (ii) Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.35. (iii) The number of pensions that the city will provide on behalf of each new hire is independent of the number of pensions it will provide on behalf of any other new hire. Determine the probability that the city will provide at most 62 pensions to the 110 new hires and their husbands. Enter your answer as a number accurate to 4 decimal places.

Consider a single firm producing an allergy medication under patent. (You can think of the quantity of allergy medication in terms of bottles of medication). The demand for allergy medication is given by ?? = 100 ? 0.1????. The marginal cost of producing allergy medication is given by ???? = 5 + 0.3???? .

a. Graphically illustrate the profit maximizing quantity of allergy medication and the price the firm charges. Though you do not have an equation, draw in an average cost curve for the monopolist in your graph.

b. Label profit in your graph. In the graph you’ve drawn, is the monopolist earning positive economic profit? Does this tell you whether the monopoly is operating in the long run or the short run?

c. Calculate and label the allocative efficient quantity of allergy medication in your graph.

d. Does the monopolist charge a markup above marginal cost? If so, calculate the size of the markup.

Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.78.

(a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 19 specimens from the seam was 4.85. (Round your answers to two decimal places.)

  ,

(b) Compute a 98% CI for true average porosity of another seam based on 13 specimens with a sample average porosity of 4.56. (Round your answers to two decimal places.)

  ,

(c) How large a sample size is necessary if the width of the 95% interval is to be 0.3? (Round your answer up to the nearest whole number.)
specimens

(d) What sample size is necessary to estimate true average porosity to within 0.25 with 99% confidence? (Round your answer up to the nearest whole number.)
specimens

STAT 14_3:

Ronit has a box with beads. The beads are opaque or transparent and available in several colors.
The probability of a random bead being red is 0.3. The probability of a bead being transparent is 0.6.
Of the red beads - the probability of a random bead being transparent is 0.5.

a. Remove 8 beads from the box at random and upon return. What is the probability that exactly two of them will be red?

b. Take beads out of the box accidentally and on return until you first remove a transparent bead
i. What is the probability of getting more than 4 beads?
ii. The first two beads taken out were not transparent. What is the probability of getting 7 beads out of the box?

c. Remove 10 beads from the box at random and upon return. What is the probability that exactly three of them will be red and transparent, two opaque and red and 5 transparent and red?

Robert Company makes bottles. The followings are the extracted information:

The Company gets a special order of 8,000 units. If the Company accepts the order, it has to incur an additional package cost $0.3 per unit.

a)  Calculate the profit /(loss) impact if the Company accepts the special order, (assume no other fixed costs are affected.) if the special order unit price is $20.

b) Advise if the Company should accept the special order quantitatively.  

Problem 13.35. A biotech manufacturing company can make test kits at a cost of $ 20.00. Each “kit” for which there is a glove demand the week it is manufactured can be sold for $ 100.00. However, due to the short life of its components, each “kit” that cannot be sold during that week has to be discarded at a cost of $ 5.00

The weekly demand for that product is a random variable with the following pmf:

1) Calculate the expected value of the demand for “kits” during one week (5 pts)

2) Calculate the variance of demand for a week (5 pts)

3) The company could manufacture 50 “kits” per shift. If they decide to go into business, how many shifts must they work (1, 2, or 3) to maximize the expected profit?

T/f: The mean absolute deviation is more sensitive to large deviations than the mean square error.

T/f: A smoothing constant of 0.1 will cause an exponential smoothing forecast to react more quickly to a sudden change than a value of 0.3 will.

T/f:An advantage of the exponential smoothing forecasting method is that more recent experience is given more weight than less recent experience.

T/f: Linear regression can be used to approximate the relationship between independent and dependent variables.

T/f:"Forecasting techniques such as moving-average, exponential smoothing, and the last-value method all represent averaged values of time-series data."

T/f: The moving-average forecasting method is a very good one when conditions remain pretty much the same over the time period being considered.

Assume you have applied for two scholarships, a Merit scholarship (M) and an Athletic scholarship (A). The probability that you receive an Athletic scholarship is 0.18. The probability of receiving both scholarships is 0.11. The probability of getting at least one of the scholarships is 0.3.

1. A horse race contains 8 horses. If you wished to bet on every possible top three finishes (first, second, and third place), how many bets would you need to place?
2. The following represents a binomial experiment. Seven plants are operated by a garment manufacturer in seven different countries. There is a 10% chance of strike at any time. It is also believed that strikes at one plant do not influence strikes at any other plant (because these are in third world countries with little access to news information.)  What is the probability of at most two plants going on strike?
3. The pH measurements of water specimens from various locations along a given river basin are normally distributed with mean 8 and standard deviation 0.3.  What is the probability that the pH measurement of a randomly selected water specimen is dangerous—less than 7.2?

Greta, an elderly investor, has a degree of risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 8.8% per year, with a SD of 23.8%. The hedge fund risk premium is estimated at 13.8% with a SD of 38.8%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual returns on the S&P 500 and the hedge fund in the same year is zero, but Greta is not fully convinced by this claim. Calculate Greta’s capital allocation using an annual correlation of 0.3